Differential and Integral Calculus, with Examples and Applications

Home , Differential calculus

TARY TREATISE

DIFFERENTIAL AND INTEGRAL

AL C CULUS,

W ITH EXA MP LES A ND AP PLICA TIONS.

B . N S. GEOEGE A . OSB OR E , ,

PROFESSOR OF MATH EMATIC S IN TH E M ASSACHU SETTS

INSTITUTE O F T ECHNOLO GY .

H HE TH CO . PUBL S ERS . A D. C , I BOSTON NEW YORK C HICA GO

1903 C OPYR IGHT 1891 , ,

B B Y GEORGE A . O S ORNE REFA E P C .

T IS k n n as a x -b k co and scien H wor , i te ded te t oo for lleges tiﬁc c ba o n m m as m s hools , is sed the ethod Of li its , the ost rigorou s and most intelligible form o f presenti ng the ﬁrst

Th e m o f m h a principles Of the s ubject . ethod li its s also the important advantage o f being a familiar method ; for it is n o w so generally introduc ed in the study o f the more ele m n ar anc ma ma c a u n ma be e t y br hes Of the ti s , th t the st de t y assumed to b e fully c onvers ant with it o n beginning the

Differential Calculus .

o mu a f n a n in The rules r for l e for di fere ti tio Chapter III .

in o n e c m i n m a x -b k in b n differ respe t fro those si il r te t oo s, ei g

x in m u n a Of x u b n an unc n e pressed ter s Of i ste d , ei g y f tio

T are u c a cab all x n o f x. hey th s dire tly ppli le to e pressio s, withou t the aid o f the u sual theorem c oncernin g a function o f a function .

A ac u n c f n a n u n fter q iri g the pro esses of di fere ti tio , the st de t

i n a V . n uc n a n a n as a Ch pter is i trod ed to the differe ti l ot tio , c onvenient abbreviation o f the c orresponding expression s by

n T n a n h a m an dvan differential c oefﬁcie ts . his ot tio s ifest a tages i n the stu dy Of the Integral Calculus an d in its

n applicatio s .

I n u n u In Chapter X . a d subseq e t pages I have introd ced for 8 Pa a n a o n n a n — c has c n rti l Differe ti ti the ot tio 7 whi h re e tly 6a: n c ome into such ge eral u s e . The chapters On Maxima and Minima h ave been pl aced

a a ca n cu as c n a n a fter the ppli tio s to rves , the o sider tio Of th t s ubject is muc h simpliﬁed by representi ng the function by

n u M x m M e ak n a Of a c . a a and n ma ma b the ordi te rve i i i y t e ,

ua a an a mm a a C a if desired, with eq l dv t ge i edi tely fter h pter III X .

u k m In a X . n a Ca c u a a n b Ch pter , I tegr l l l s, I h ve t e the pro le o f n n M m n n a o f a an a a as a b ﬁ di g the o e t of I erti pl e re , etter illustration o f double i ntegration than that Of ﬁnding the

The u n m a c m n n area itself. st de t ore re dily o prehe ds the i de

n Of a: an d in ub n a pendent variatio y the do le i tegr l,

2 x dx d an in doc d . ( yﬁ y, th y

A a o f a XII n a Ca cu u are d few p ges Ch pter , I tegr l l l s, evoted to a description o f the Hyperbolic Functions together with

f n a and a c m a n ma c o r their dif ere ti ls, o p riso is de with the

responding Circular Functions . E A . O B R . G . S O N

BO TON 1895. S , NT C ONTE S.

DIFFERENTIAL CALCULUS.

CHAPTER I .

FUNCTIONS.

Deﬁnitio n and Classiﬁcatio n o f Functio ns

No tatio n Of Emotio n Exam les s . p

CHAPTER II .

DI ERENTIA E I IE T FF L CO FF C N .

mi In r m n Li t . c e e t - m l Diff r n i l fﬁ i nt . Ex a es 8 10 . e e t a Co e c e p

I CHAPTER I I. E DIFFER NTIA TION.

l r i Fun tio ns Exam e —2 Differentiatio n o f A geb a c c . pl s 10 1 Diff renti ti n o f Lo arithmic and Ex o nential Functio e a o g p ns . Examples iffer n i io n Of Tri o no metric Fun ctio ns Exam les D e t at g . p . iff r f n r ri n m ri F n i n Ex D e entiatio n o I ve se T go o et c u ct o s . — amples 32 3 7 21 22 Differentiatio n o f In erse Functio n and Functio n Of a , . v Ex m l 3 -40 Functio n . a p es 7

CHAPTER IV. E I E DI ERENTIATI N SUCC S S V FF O .

23 24 . Deﬁniti n nd N t i n , o a o at o

25 . The h i r fﬁ i Ex m l nt D ffe ential Co e c ent . a p e s ’ 26 . Leibnitz Th r m Ex m l s eo e . a p e s vi TE T CON N S .

CHAPTER V.

DI EREN I FF T ALS. A M E.

Diﬁ erentials as relate d t o Differential Co efﬁcients Differentiatio n b y Differentials Successi e Differentials Ex am les v . p

I CHAPTER V .

IMP I IT UN TI NS L C F C O .

iff r nti tio n o f Im licit Fun i E - D e e a p ct o ns . xamples 52 54

CHAPTER VII .

EXPA SI N OF UN TI S N O F C ON .

ri ’ Th o r m Exam l Maclau n s e e . p es ’ Ex m l Taylo r s Th eo re m. a p es ’ Rigo ro us Pro o f Of Taylo r s Theo rem ’ ’ Remainder in Taylo r s and Maclaurin s Th eo rems

CHAPTER VIII .

I DETERMINATE RMS N FO .

u f Fra tio n Limiting Val e o 3. c

Evaluatio n o f 3. Examples

E m le E aluatio n o f 0 co co 00 . xa s v g, , p

rm Exam les Evaluatio n o f Exp o nential Fo s . p

CHAPTER IX. N PA RTIAL DIFFERENTIATIO .

tial Differential C o efﬁ cients Of First Order . Exam 9 6 . Par 5 , 0

— Hi h er rd r Ex m Differential Co efﬁcients o f O e s . a 61 63 . Partial g — ples 80 82 Fun tio ns Of Se eral Variables To ta l Diﬁ erential Of c v . Examples Ex m le s Co nditio n fo r an Exact Differential . a p n tio ns Differentiatio n Of Implicit Fu c . ’ Taylo r s Th e o rem fo r Several V ariables S ii CONTENT . v

CHAPTER X .

CHANGE OF VARIABLES IN DIFFERENTIAL E COEFFICI NTS.

Ch angi ng fro m so to y Ch anging fro m y to z h an in fro m to z Ex am les C g g ac . p

CHAPTER XI .

REPRESENTATI OF VA RI US URVES ON O C .

4- 85 Rectan ul ar Co -o rdinates 7 . g — - 86 93 . Po lar Co o rdinates

CHAPTER XII .

D E R E TAN EN D IR TI F U V T AN RMA . C ON O C . G NO L

A SYMPTOTES.

- i 4 97 Direct o n o f urve ub tan ent and ubno rmal . 9 . C . S g S Example s

98 Differential o efﬁ i n h A r , C c e t o f t e c

99 E uatio n o f th e Tan ent and No rmal Exam les . q g . p 100- 106 A s m to tes Exam les . y p . p

CHAPTER XIII .

D E N IR TI OF URVA TURE. P I TS OF I F EX I C O C O N N L ON.

10 —109 Directio n Of ur ur 7 . C vat e

110 . Po ints o f Inﬁ exi o n Ex am l . p es

CHAPTER XIV.

URVATURE. IR E OF URVATURE EV C C CL C . OLUTE A ND INV UTE OL . — 111 113 . Deﬁnitio n o f Curvature Unifo rm and Variable

114 115 . Radius Of Curvatu re Ex am le , . p s 116 n r . C e t e Of C urvature — 117 12 1 . E o lute and In o lute Ex am l v v . p es CO TE T viii N N S.

CHAPTER XV.

RDER OF N TA T. S U ATIN IR E O CO C O C L G C CL .

Co nsecutive C o mmo n Po ints Os culating Curv es

A n alytical Co nditio ns fo r C o ntact

Osculatin Circle Exam l g . p es

CHAPTER XVI .

ENVELOPES .

Series o f Deﬁni i Curves . t o n o f Envelo pe Equatio n Of Envelo pe E o lute the En elo e Of No rmals Ex m l v , v p . a p es

T CHAP ER XVII .

SI GU AR P INTS OF URVES N L O C .

Multiple Po ints

Po ints Of s cul i n us O at o . C ps n u Co j gate Po ints . Examples

CHAPTER XVIII .

MA XIMA AND MINIMA OF FUNCTIONS OF ONE I DEPENDENT VA RIAB E N L .

- i i M xi m nd Minim d 4 Deﬁnitio n . C o nd t o ns fo r a a a a e 1 5 149 .

rived fro m Curves 153 157

’ 150 o nditio s o r Maxim nd Minima b Ta lo r s , 151. C n f a a y y Ex m Th eo rem. a ple s Problems in Maxima and Minima

CHAPTER XIX .

MA X IMA AND MINIMA OF FUNCTIONS OF SEVERAL E E T A RIAB E INDEP ND N V L S .

ditio ns fo r Maxima and Minima b D ﬁnitio n . Co n 155 . e y ’ l r h o r m Exam les 165-1 1 Tay o s T e e . p 7 C T ON EN TS .

INTEGR L C L L A A CU US.

CHAPTER I .

E EME TA R RMS OF INTE RATI L N Y FO G ON.

i i In r Deﬁn t o n o f teg atio n . Elementary Principles Fundamental Integrals Derivatio n and A licatio n o f Fundame ntal Fo rmulae pp . Examples

CHAPTER II .

E F ATI NA A INT GRATION O R O L FR CTIONS .

Preliminar O eratio n Facto rs Of Deno minato r y p . I Exam l s Cas e . p e II Exam le s Case . p x m l C as e III. E a p es I Ex m l Case V . a p es

CHAPTER III .

A N B Y RATI A IZ AT INTEGR TIO ON L ION.

Ex m Fractio nal Po wers Of x and o f a bx. a ples i r 2 Exam l Fract o nal Po we s Of a 523 . p es

i i u i n Exam les Integrat o n by Sub st t t o . p

I CHAPTER V.

INTE A T INTE RA TI N B Y U ES I E GR TION BY PA R S. G O S CC S V EDU TI N R C O .

Ex m l 2 1 Inte ratio n b Parts . a es . g y p 22-24 F rmul Redu ti n Exam les . o ae o f c o . p

CHAPTER V.

I N METRI INTE RA S TR GO O C G L .

m n 25—27 Inte ratio n o f tann o f s ecn dx o f tan x s ec x dx . g x doc ; x ; . Exa mples r inm ’l Exam les Integ atio n o f s x co s x dx. p X NTE T CO N S.

PAGES.

Tri o no metric transfo rmed into Al ebraic Int r g , g , eg als . Examples

Tri o n o metri c Fo rmulae o f Reductio n E g . xamples dz dc” Inte ratio n o f and g , a + b s 1n x a + b c o s x “ “ o f e s in m; dx and 6 co s mdoc u .

226-229

I CHAPTER V .

I TEGRA S FOR RE ERE N L F NCE.

3 — 2 / z 2 2 — 36 . Inte ra s co ntainin v a ao x ac :l: a ziz ax x 0 230 285 g l g b .

CHAPTER VII . MM INTE RATI N A S A SU ATI . DE I ITE I TE RA S G O ON F N N G L .

-40 Inte ratio n th e Summatio n Of an Infinit — 37 . g , e Series 236 2 40 — D finitio n o f Defin ite Inte ral Ex m l — 41 43 e . a s 4 44 . g p e 2 0 2

CHAPTER VIII .

ATI N OF INTE RATI N TO P A E VES A PPLIC O G O L N CUR . I A TI N TO E TAIN A PPL C O C R VOLUMES .

- f Curves Exam les 44 47 . A reas o . p m l 4 49 Len th s o f Curves . Exa es 8 . , g p i n Ex m 50 5 1 . Surfaces Of Revo lut o . a es , pl Exam l s h r o lumes . e 52 . Ot e V p

R CHAPTE IX.

I E RA SUCCESS V INTEG TION.

- 53 56 Double and Tri le Inte rals . Exam es 258 260 . p g pl

CHAPTER X. DOUBLE INTEGRATION A PPLIED TO PLANE A REA S E A AND MOMENT OF IN RTI .

— - in Ex 60 Do uble Inte ratio n . Rectan u ar C O o rd ate s . 57 . g g l amples — ’ - Ex m l Do ub le In te rati o n . I Olar C o o rdinate s . a es 6 1 6 3 . g p xi CONTENTS.

CHAPTER XI .

D ID SURFA CE AN VOLUME OF ANY SOL . RT A S ,

6 4 65 Are a o f an Surface . Ex am les , . y p m lid Ex m l 66 6 Vo lu e o f an So . a s , 7 . y p e

CHAPTER XII .

N TI N D E ID AND H PERB I U S. I PI Y OL C F C O CYCLO , CYCLO , D R E I TRI I E A N F A U . HYPOCYCLOI . N NS C QU TIO O C V

— Deﬁnitio ns Of H erb o lic nd n r H erb o ic 69 7 1 . a I e s e yp , v yp l , Functio ns 2 74—276

2 3 Differentiatio n o f H erb o lic Functio n In ers e H er 7 7 . s . , yp v yp b o lic Fun ctio ns as Integrals 74 75 H erb o lic Functio n s and th e H erb o la Ex rcises , . yp yp . e 6—82 E uatio and Pro ertie s o f th e C clo id 7 . q n p y — 83 89 . E uatio ns and Pro er ies o f h E i l id nd H q p t t e p cyc o , a ypo — cyclo id 284 288 — r E u i l 9 93 Int insic at o n o f a Cur e and o f its E o ute . 0 . q v v Examples 289- 292

TI L L DIFFEREN A CA CULUS.

H PTER I C A .

N I NS F U C T O .

e niti o n o a Functio n . n a u o ne a ab 1 . D ﬁ f Whe the v l e Of v ri le uan SO n u n a o f an a an c an q tity depe ds po th t other, th t y h ge i n a uc a c n n c an i n m the l tter prod es orrespo di g h ge the for er,

o m a b e a un cti o n a the f r er is s id to f Of the l tter .

Fo r xam a a a u a a unc n S e ple, the re Of sq re is f tio Of its ide ; um a a unc n o f a u n the vol e of sphere is f tio its r di s ; the si e, c n and an n are un c n Of an x n osi e, t ge t f tio s the gle ; the e pressio s

2 2 / x lo 53 x , g ( 4 x are function s Of at . A uan ma be a unc n o f o r m a q tity y f tio two ore v riables . Fo r xam a a o f a c an a un n f e ple, the re re t gle is f ctio o two adj acent sides ; either side Of a right triangle is a func tion of the two other sides ; the volume o f a rectang ular parallelo a un n o f m n n piped is f ctio its three di e sio s . The expressions

2 2 x x + y + y 2 are unc n a: an f tio s Of d y. The expression s

2 w z z lo w — z y + y + n g ( + y . z ) are unc o n w and z f ti s Of , y, .

2 . e en ent and In e en ent r i l D p d d p d Va a b es . If y is a function at as i n u a n of , the eq tio s

= tan 4 x y , 2 DI ERE TIA A FF N L C LC UL Us .

x c a i n e en ent a ab and n nt is lled the d p d v ri le, y the dep e de variable .

It n a n a un c n x x ma b e is evide t th t whe ever y is f tio of , y a a as a unc o n Of an d o n lso reg rded f ti g, the p sition s o f depende t an d n n n a ab T i depe de t v ri les reversed . hu s fro m the prec eding ua n eq tio s, —l x x/ x tan x o . g, 7} g, l g , y

In u a n n n m an a ab as eq tio s i volvi g ore th two v ri les,

x x — = 0 + y , o n e mu be a as n n a ab and st reg rded the depe de t v ri le, the

n n n ar ab others as i depe de t v i les .

li i a n m li it un ti n n Ex c t d I c F c o ns . n o e ua 3 . p p Whe q tity is x c in rm o f an m a e pressed dire tly te s other, the for er is s id to n n b e an exp li cit fu ctio of the latter .

Fo r xam an x c unc n x i n u a n e ple, y is e pli it f tio of the eq tio s,

g = ﬁ + 2 n

When the relation between y and x is given by an equation n n n u an bu t no t o nc to c o tai i g these q tities, s lved with refere e g, be an im licit un c n x as in ua n y is said to p f tio of , the eq tio s, = y + l o g g x.

m m as i n o f u a n can So eti es, the ﬁrst these eq tio s, we solve

u a o n n c and u c an unc n the eq ti with refere e to y, th s h ge the f tio m m x c T u ﬁnd m u a n fro i plicit to e pli it . h s we fro this eq tio , y

4 A l ebra ic a nd Tra nscen enta l F uncti o ns . A n a l ebra ic . g d g u nc n o ne a n n a n o f a t n f tio is th t i volves o ly the Oper tio s ddi io ,

ub ac n mu ca n n n u n and u n s tr tio , ltipli tio , divisio , i vol tio evol tio

All u nc o n are c a tr with con stant expon ents . other f ti s lled a n n n nc u n lo a rithmic ex o nenti a l tri o s cendenta l fu ctio s, i l di g g , p , g n tri and i nverse tri o no metri c unc n . o me c, g , f tio s FUNCTIONS. 3

h m x x No ta tio n o F unctio ns . T e b F 5 . f sy ols ( ) , f ( ) , k r u no un n f x an d a e c o . T u the li e, sed to de te f tio s h s instead o f y is a fu nction of we may write

y =f (w) o r

A functional symbol occu rring more than once i n the s ame problem o r dis cu ssion is u nderstoo d to de n ote the s ame fu n c

n o r o a n a u a f n u an tio per tio , ltho gh pplied to di fere t q tities . T u h s, if (1)

2 - 2 2 5 9 1 6 . f < > 4 . f ( )

In all these Cx pression s f ( ) den otes the s ame operation as n b a a o n Of u a n u an deﬁ ed y th t is, the Oper ti sq ri g the q tity and a n 5 u ddi g to the res lt . Th e followin g examples will further illu strate the n otation n o f fun ctio s .

EX A M P L ES .

= 3 — 2 — h 1 . x 2 x x 7 x 6 sh o w t at If f ( ) + , 3 = 30 2 = 4 o = 6 l = o f ( ) 2 f ( ) 9 f ( ) 7 f ( ) 2 3 2 w — 2 = 2 w —13 x 21 w f ( f ( ) + . — — 3 2 h 7 ) x + 2 h 2 — h 7 h + 6 .

4 3 2 n 2 — 1 7 — 6 1 Sh at Give fl (y) g y + , f2 (y) y y + ; ow th 2 = f1 ( ) f2 (

a S a If f ( ) 311, how th t

1 1 + a b 4 DI EREN TIA A U Us FF L C LC L .

— 4 . If 1 ) (m Show that 2 0 M ) . Mm) , m + 2 m —2

— — — 5 . d x b x c a If ) ( ) ( ) , show th t

a + b + c 2

T ” b — 0 ¢ ) ﬁg?) —l L )

o a 6 . If sh w th t

1 x F x = l o S o a 7 . If ( ) g , h w th t 1 x

F Q’) F (z)

x lo x M o a 8 . If f ( ) g ( + ) , sh w th t

2 2f ("3) = f (2 “7 3 = ~ 3f (m) f <4 n 3 x) . — G n x c o s x 1 Sin x a 9 . ive d( ) ; show th t

“ “ MCI/ i

3 3 3 x z x z 3 x z S o w a 10 . If f ( , g, ) g g , h th t

x Z N f ( : y) )f ( p , q, M ) :

x rz where L p gy ,

z TIE s y q ,

6 DI ERE TIA C A U s FF N L L C L U .

A ne a ti ve n c m n a ecrement a a ecrea se in g i re e t is d ; th t is, d u val e . 2 Fo r xam c a n x 10 as b i n x e ple, lli g , efore, g

Ax —2 n A if , the g

‘ 3 er nti a l e i i In = D e Co c ent. x 8 . ua n ifi fi the eq tio g , if we u x t o a a a To fix a n n u n s ppose v ry, g will v ry lso . the tte tio po a n a u o f x u s u = n r defi ite v l e , let s ppose x 10 a d therefo e

: 100 an d u s n u a a n n c m n g , let i q ire wh t dditio or i re e t will Cal b e produ c ed in g by a c ertain incremen t assign ed to x. c ul ating the valu es o f Ag c o rrespon ding to different valu es o f Ax ﬁnd u as in o n ab , we res lts the foll wi g t le

i Th e third c olumn gives the value o f the rat o between the in crements of x an d o f g . ab a as Ax m n and It appears from the t le th t, di i ishes

A a m n and a ac z . approaches zero, g lso di i ishes ppro hes ero

n a o f a ac n z a The a diminishes but i ste d ppro hi g ero, p r tio 3 , pro ach e s 20 as its limit .

‘ This limit o f g is c alled the diﬁerenti a l co eﬁicient Of g with A E I IE T DIFFEREN TI L CO FF C N . 7

dg c x and IS n b In ca n x 10 respe t to , de oted y this se, whe , dx da 2 0 . dz dg be n c a no t n as a ac n It is to oti ed th t is here deﬁ ed fr tio , dx but as a single symbol denoting the limit of the frac tion Ag dg The studen t will ﬁnd as he advances that has many A x dx

o f an na ac n and a Of the properties ordi ry fr tio , Ch pter V. shows how it may be regarded as such .

u c n u an o n n m 9 . e u ca a u Witho t restri ti g o rselves to y eri l v l e, d a 2 we may obtain from the equation g x thus dx

2 a n = x Ax = h and new H vi g g , let , let the value o f g be denoted by therefore

n b Ax h Dividi g y , gives

Ay 2 x h. Ax

The m o f n it a ac z 2 x li it this, whe ppro hes ero, is . Hence

da 2 x. dx

In m the s a e w ay the differential c oefﬁcients o f other given unc n ma be un f tio s y fo d .

Fo r xam ﬁnd e ple, m ua n 31fro the eq tio ,

A9) h ,

8 : 2 x 7 g ( 0 + 1 .

’ 3 3 2 2 3 A : 2 x h 2 x : 2 3 x h 3 xh l g g g ( ) ( + f h ) . 8 DI ERE TIA A FF N L C LC UL Us .

Dividing by Ax it gives A y 2 2 3 x 3 xh ( + W . A x )

2 The m o f IS 6 x as h a a z li it , ppro ches ero .

— 2 31 6 5 .

Take fo r another example = Ax h.

\/x

9 g \/x + h A x h

The limit o f this takes the in determinate form But by r tio nalizin num a a a g the er tor, we h ve

Ag 1 A x

53—9 1 The limit o f Aw 2 \/x

dg 1 a I th t s , div 2 xfi

1 Genera l e ni tio n o D erential Co e ici ent. 0 . D fi f ifi fi

In n a ge er l, if g

' = ! h y TO ) .

' < h < m M y y MM ) M ) .

' A x h _ ?! (I) x h (I) x limit o f as h approaches zero . gx h

m n n The difierential co effici ent o f a function ay the b e defi ed DI ERENTIAL E I IE T FF CO FF C N . 9 as the limiting va lue of the r a ti o of the i ncrement of the functi o n

o the i n rement o the vari a ble as these i ncrements a r t c f , pp o a ch T a ff n a n f un n zero . c c o c x h t is, the di ere ti l oeffi ie t the f tio ¢ ( ) c x with respe t to , is — 9 “W h ih the limit o i iz as h is in definitely diminished . The ff n a c c n m m ca the eri va tive di ere ti l oeffi ie t is so eti es lled d .

— 4 m n In A rt . 9 b e un ca u NOTE . will fo d a geo etri l ill stratio f f n o the differential c oe ﬁcie t .

EX A M P L ES.

F n f A c o rt . 9 n ollowi g the pro ess , derive the followi g ferential c oefﬁcients d J 2 v 1 . 3 x 2 x. 6 x 2 g . dx

d — 3 = x 4 x 1 7 g ( + . di

1 dg 1 “4 ° if 2 5 dx 5

a dg 2 a J “ 5 ' y 3 ? dx x — x a dg 2 a J G’ y x + a I dg Sx W7 dx 2

dg x

2 dx \/x —2 dg 1 dx

dg 1 H PT C A ER III.

R EN N DIFF E TIA TIO .

1 1 The c n n f n a c fic n a . pro ess of fi di g the di fere ti l oef ie t Of n u n n a n Th e x am i n give f ctio is c lled difierenti a tio . e ples the p rec edin g c hapter are introduc ed to illu strate the mean ing o f ff n a c fic n but m n a m the di ere ti l oef ie t, this ele e t ry ethod of

ff n a o n u r u di ere ti ti is too tedio s for gen e al se . Diﬂe rentiatio n IS more readily performed by the application

c a n n a u m e x r o rmu a c a b b . Of ert i ge er l r les, whi h y e p essed y f l e

In r mu a u and v n o a ri a ble uan unc these fo l e will de te v q tities, f

d ns n n x and c an n co ta nt u a . tio s of ; , q tities It is frequ ently c on venient to write the diﬁ erential c o efﬁcien t o f a qu antity d u n a , i ste d of dx

Thu s the differential co efﬁc ien t o f ( u + v) is more c o n ven iently written

l u v a an ( ) , r ther th alx

- Fo rmulae o r D eren ti a tio n o A l ebra i c F u nctio ns . 12 . f iﬁ f g

dx d w da dv dx

d du dv Q u 21 d dx DIFFEREN TIA TION .

These formul ae express the followin g gen eral rules o f dif fe renti atio n

The d erenti al co e ici ent o I . ifi fi f a vari a ble with resp ect to i tself is unity. . ’ The d erentia l co e i i n . c e t o ns a n s II ifi ff f a co t t i zero .

II The d erenti al co e ici ent o h u I . iﬁ ﬁ f t e s m of two va ri a bles i s

su o thei r d erentia l co e i ci ents the m f iﬁ ﬁ .

‘ The d er enti a l co e i ci ent o the ro u t o two va ri les IV. ifi fi f p d c f a b is the sum of the p ro ducts of ea ch va ri a ble by the difierentia l

o the o ther co eﬁici ent f .

’ The d er enti al co e cient o the ro u ct o a co n sta nt a n d V. iﬁ fﬁ f p d f a vari a ble i s the p ro duct of the co nsta nt a nd the d izferentia l co

i n th a i a eﬁ ci e t of e v r ble.

‘ ' I The n ti i ra ti n i s he d e n i V . difiere a l co efi ci ent of a f c o t ifi re t a l co efiic ient of the numera to r multip li ed by the deno mi na to r mi nus

' the diﬁerentia l co eﬁicien t of the den o mi na to r multip lied by the

’ n umera to r this d erence bei n ivi e b the s ua re o the , iﬁ g d d d y q f

eno i n a t d m o r . ‘ II Th wer o a va r ia ble i s V . e difierenti a l co efiicient of a ny p o f the ro uct o the ex o nent the o wer wi th ex o nent imi nishe p d f p , p p d d ‘ b 1 a nd the d erenti a l o e icient o the vari a ble. y , ifi c fi f

eri ti lae 1 . va o n o F o r 3 D f mu .

P r o o o T o m m n n o f f f I . his f llows i mediately fro the deﬁ itio 5 9 a n ﬁi i n n 5 1 m 95 1 t F r . a c o e c e . o c differe ti l si e , its li it Ax dx

ro A n n u n u P of of II . c o sta t is a q a tity whose val e does no n t vary . He ce 12 DI ER E TIA CA L C UL FF N L US.

A0 0 and

therefore its limit dc dx

P ro o = o III. Let u v and f f g + , suppose that when x is ' ' ' c an n x + h u an d v b c o m u an d v n h ged i to , g, , e e g , , ; the

r 1 y : “+

' ' therefore g g u h . u q,

a A Au A th t is , g v .

Divide by Ax ; then

g Au Av A + Ax Ax Ax

No w u Ax to m n and a ac z n s ppose di i ish ppro h ero, a d we a fo r m o f ac n h ve, the li its these fr tio s,

dg da d v + dx dx dx

i n w e ub u u v If this s stit te for g, , we have

du dv

It is evident that the s ame pr o o f would apply to any nu mb Of a ab nn c b u o r m nu n er v ri les co e ted y pl s i s sig s . We should then have

d dw ’ i f i q (u i v i w i u i i jCU dx dx dx

Let uv n P ro of of IV. g ; the

r b t : “u y ,

' ' ' ' g g u v u v ( u u ) v + u (v v) ;

' Ag v A u uAv .

14 DI ERE TIA A U US FF N L C LC L .

L P ro of of VI. et

f u u . v ' h ) (v ) therefore g g ;

a th t is,

All Ax

No w u Ax m n a z and n c n s ppose to di i ish tow rds ero, , oti i g ' a m o f v v w e a th t the li it is , h ve

dv dx

m I I . derive V . fro V E

therefore gv u .

I By V.

therefore 15 DIFFEREN TIA TION .

F irst u n b e a n . o o . P ro f f VII , s ppose to positive i teger

u" Let g ,

n d u a , g

'n ‘ 1 2 a A A u u u th t is, g (

A — — A“ y [ ri —2 171 3 2 n l u u u u u u A x Ax

' No w A x m n n u b n m o f u ac let di i ish ; the , ei g the li it , e h o f the n terms within the parenthesis becomes therefo re

’ l — d ll / fy n l dx dx

eco n u n b e a o ac n S d, s ppose to p sitive fr tio

P L t u ﬁ e g , then g? up ; d d thero fore

S n II b e u n e x But we have already how V . to tr e whe the pon ent is a positive integer ; hen c e we may apply it to eac h u n T member o f this eq atio . his g ives

dg 1 du 2 up qy P 2 dx dd:

—1 dg p u p d u therefore ‘ l dx n dx

Sub u n fo r us stit ti g g, , gives — P l dg p u d u p g- I du u ’ — dx g p dx q dx a g

II i n . c a which shows V to be tru e n this c ase also . He e th t o rmu a a an a u n n a f l pplies to y positive v l e of , whether i tegr l o r a n fr ctio al . 1 6 DI ERE TIA A U S FF N L C LC L U .

suppose n to be n egative and equ al to 1 y u u

(u ) n, —m—l b VI. mu y , 2 m d x u

II u n n . a He c e V is ivers lly tru e .

EX A M P L ES .

Differentiate the following fun ction s

4 1 (13 . . y

u an are ua f n a c ﬁc n If two q tities eq l, their di fere ti l oef ie ts u u a nc m st be eq l . He e

4 (x ) d x dx

b u n u x and n 4 a a . u If we pply VII , s stit ti g , we h ve

dx d 4 = 3 3 4 x 4 x b I .

13 — s .A n dx

= 4 3 2 3 x 4 x . . g + l dy 4 f 4 i 3 £1. <3 x + = (3 m) + (4 x dx dx dx dx

4 b U L mak n i t 3 x an d v y , i g

d 4 S b v . (3 5 ) g n, y , dm i

3 3 : 3 4 x 12 x .

2 ” d 3 d 3 = - = x Si mi a 4 x = 4 x 4 3 x 12 . l rly, ( ) ( dx dx

7 3 2 3 2 A : 12 2 + 12 x l 2 (x + x ) . 3x DIFFERENTIA TION 17

d g d

d i t o oa b . ( ) g , y VII

d 2 O b ( ) ) y dx

dg d x 3 2 dx dx x + 3

Applying making

2 x 3 an d v = x 3 w e h v + + , a e

2 d d x 3 i dx

2 ( 96 4-3 )

2 (90 +

— — 2 dg _ 3 6 x x Z — dx (x i 18 DI ERE TIA A L FF N L C LC U US.

dg d h dx dx

a II mak n If we pply V . , i g

2 u = x + 2 and n w e have g,

2 2 —1 2 2 I d i’ x — x Z — x

4 ” 3

gig 4 x d“?

— g x .

d y 3 i = n“ 1) (90 s ol dx i

a m k . a n we pply IV , i g

= 2 3 u x + 1 and v = (x we have

i i [ 69 + 1) (ar o ]

2 i i i E (00 + o o w> + w) e n 1) di , w

8 i d k —

du % . 3 — A l ) (x x) x) 2 x dm

2 2 — 4 — 2 gx + 1 ) § 3 x x ) 7 x 2 x 1 ° 3 ; 3 t 2 (x x) 2 (x a n DI ERE TIA TI FF N ON . 19

5 3 dv 4 2 x 1 16 x 1 x l 2 8 . ( ) ( ) ( ) ( x dx

— § I § x 5 x + 2 x + § x

— e + 2 ﬁ —3 o

2 x%

n 12 . Give

5 5 4 3 2 2 3 4 5 (a x) = a + 5 a x 10 a x + 10 a x + 5 d x + x ;

4 derive by differentiation the expansion o f (a x)

? ” ‘ 1 sum 1 2 x 3 x ” nx derive the of the series .

’ l nx’t f n l ) x” + 1 . A ns . (ac

_d_y_

doc l u r 2 ( k /1 m m

x” big (1 x) " dx (1

2 = 1 —2 x 3 x 16 . g ( + fig

= 17 . g (1 dx 20 DI ERE TIA A U US FF N L C LC L .

’ ’ (d + 2 a x 12 x ) dx

— 5 3 x) (a

“ ‘ ’ 2 15 x a —3 x a a 2 ax —23 x ( ) ( ( + ) . dx

y : (a (b l — “4 " [m (b+ x) EiZ

“ (a x) (b x) ” dg dz

gig dx ( 1

l + x

do:

(3 w ok/w

S x L

“ S I S — 2 2 x ( 1 x

mam 1) i

i 9 . i 2 ) .

22 DI ERE TIA L CAL C UL Us FF N .

No w n z Inc as s n n a e whe re e i deﬁ itely, we h v

limit o f

T s uan u ua n b 6 so a hi q tity is s lly de oted y , th t 1 1 1 e —1

The value o f C c an b e eas ily calculated to any desired numb er o f decimals by c omputing the values o f the succes sive

F ma a h a u a i n terms o f this s eries . o r seven deci l pl ces t e c lc l t o i as f s s ollow ,

. 5

. 166666667

. 041666667 008333333

. 00 1388889

. 000198413

. 000024802

. 0000027 56 00000027 6 000000025 000000002

h a u fo r d fe ent values o f 2 By calculating t e v l e if r ,

T us we may verify its limit . h ( 1 “ ( 1 i ) ( 1 TIA TI 23 DIFFEREN ON.

erivatio n o F o rmulae . 1 6 . D f

P ro o o I. Let u f f VII g log a ,

' n u Au the g log , ( ) ,

A l o u Au u lo g g , ( ) log a g ,

— 10 1 lo 1 e. g .

n b Ax Dividi g y , Av— 1o 1 e. Aw

ac z Au at am m a ac No w if Ax appro h ero, the s e ti e ppro hes

zero ; then the limit o f is the s ame as the limit

2 f i n n n But Art . 15 o f as i i creases i deﬁ itely. n we

n have already found the latter limit to b e e . He c e we have alu

dg dx lo e ” ga 7 dx a

T a c a ca o f . n a : . e . Pro of of IX his is spe i l se VIII , whe In this l o e lo e 1 . g a g e — NOTE Logarithms to base 6 are called Nap i erian loga hm rit s . a n n o ba c Na an Here fter, whe se is spe iﬁed, pieri n lo garithms are to b e u derstood .

T a l o u u . h t is g log ,

P ro of of X. L “ et g a .

Tak n a m o f ac m mb a e i g the log rith e h e er, we h v

log g u lo g a ; dy

th f g b X . ere ore y I , : 31 DI ERE TIA A 24 FF N L C LC UL US.

“ Mu n b : a a ltiplyi g y g , we h ve

1 / 9—y u g f = 0 . 10 g a, (L dx dx

T a c a a : P ro o o . c o f X . a 6 f f XI his is spe i l se , where .

II L ” et u . P ro of of X . g

k n a m o f ac m mb Ta i g the log rith e h e er, we have

log g v lo g u ;

dg u v d d” dx b . therefore y IX , log u y u

Mu n u ” a ltiplyi g by g , we h ve

+ log u -u”

EXA M P

1 .

= l x o x. 2 . g g

= " l x g x o g .

= / — 2 l o \ I 7 . 2 g g 1 — x

x 2 3 e 1 —3 x —x ( ) .

g \/x

g lo g (63 DI ERE TIA TI FF N ON. 25

” ‘ x — 4 x x : 2 x y ( 3 ) e c . g (

d 5 = 8 g- = 1O 1° 5 37 + x M Q y g ( 3 dx 5 x + x 1 - = where M . lo gme . 434294 1 10 g‘ 0 d a 10 g 5 fig 4 dx

What is the result o f differentiating bo th members o f e ach o f the three follo wing e quations ?

{ 13 . I

l ’ A ns 1 —x + x L i-x

3 l 1 1 4 . o g + 3 + § g 7

A ns

15 .

A ns .

1 6 . g

1 — 7 . g lo g (x 2)

dg Va 18 . lo g g M . 26 I ERE TIA A U S D FF N L C LC L U .

dg lo g x — ’ dx ( 1 x)

d 1 5 g J , 3 x + 6 x we dx 2

2 3 x 6 zv “ 3 “ g = e + g = ax e 2 a a dx

d lo lo x lo x lo lo x l g g ( g ) g g g ( g ) o g x. dx x

— 2 g = lo g (x 3 V x

= 26 . g log

— ITF; 2

27 .

g_g 2 x y = 10 g d“? 2 2 2 2 x + a V x + b

2 g x 1 = lo d y e ‘ 2 dx x + x + 1

2 ’ 4 ° g (e' (e + 2 e + 1 )

d _y — ( 1 lo g x) dx

d?! + 10 9; IA TI 27 DIFFERENT ON .

——j l o g x) . 33 . g giG

dg _ 1

l x o g z 35 . y .

_1_ 1 x 0 “ 36 . g .

’ 33 e ” e . . 6 e 37 y .

z x” x x' 1 l e . e o x . 38 . g ( g )

x " 2 (La x lo x lo x 39 . y . g g ( g )

Fo rmula: o r D erentiati o n o Tri o no metric Functio ns . 1 7 . f iﬁ f g In the follo win g formul ae the an gle u is supposed to be ex in m pressed Circular eas ure .

sin u co s u

c o s u

tan u s ec u

an u XVII . s ec u se c u t

II u ° XVI . c osec

in XIX . vers u s u 28 DI ERE TIA A U FF N L C L C L US.

18 . eriva tio n o Fo rmul D f ae.

P ro o o . Le t sin i t f f XIII g ,

' = then g sin ( u Au) ; therefore Ag sin (u Au)

But m T n m fro rigo o etry,

in — s A si n B 2 sin %(A B ) c o s %(A

ub u A u Au and B u If we s stit te ,

$ 71 w e have Ag 2 co s . 2

2 A“ Henc e co s u + A u Ax 2

No w n Ax a ac z Au k i a a whe ppro hes ero, li ew se ppro ches z and as A u in c cu a m a u m o f ero, is ir l r e s re, the li it

un is ity.

Henc e

T ma be b ub u n i P ro of of XIV. his y derived y s stit ti g n

17 X u u . III . for , 2

Then

S 30 DIFFERENTIAL CAL CUL U .

EX A M P L ES.

= x —Sin 2 x sinx = . 2 co s 2 x co s y sin 2 x co s x gg

d z g _ ’ = x _ 10 tan 5 x s ec 5 x . y tan 5 . dx

d —v= 2 = x x tan 27. g ta.n dx

_dg__ dx

Qy— = M = s in x + co s x. s ec x dx

d ’ 3 v_ ’ ’ — = sin x 3 co s x Sin x . y = 8 m x co s x. ( ) dx

dy —= 0 0 8 2 x co s (x a ) ° dx

s1n ’ Q 8 111 2 a co sec (a x) dx sm (a + x)

d?! s _ : 2 tan x y tan x a

’ ’ " 5—1 2 x a sm x b co s x 5 n (a b) sm ( ) dg x. lo sm x co t 12 . g g dx

= lo tan x . 13 . g g dx siu 2 x

lo sec x. 14 . g g DIFFERENTIA TION 31

= ers 15 . g v vers

S I sin 2 x. S

S I “ e sin x . S

S I 17 . g co s x lo S g

S ” I h l in nx siu x. n sin x s n n 1 x . 18 . g s i ( + ) S

“‘ 1 S mn s in nx c o s m n ) x I S

S 2 2 x l I O 0 . g + o g co s S 1 + tan x

S I 21 . lo a x g g t se c . S

22 . lo g g co se c x. dx

dg_ a b 2 0 3 . y lo g 2 z ’ z b siu x dx a co s x b s in x

S I co s 4 x . S

In each o f the foll owing pairs o f equations derive differen tiatio n e ach o f the two equations from the other :

25 . sin 2 x 2 s u x o s x i c ,

z ’ s 2 x u x co co s x si .

2 tan a 1 2 x 26 . 8 11 ; 1 tan x

z 1 tan x co s 2 x . z 1 tan x 3 in x 3 Sin x 4 sin x s 3 ,

3 c o s 3 x 4 co s x Bo o s x .

3 3 u 4 x 4 u x c o s x 4 co s x sin x si si ,

2 z in x c o s 4 x 1 8 s c o s x .

x sin mx o s nx c o s mx Sin nx si n (m n ) c ,

in c o s (m n ) x c o s mx c o s nx si n mx s nx.

3 5 7 x x x SID £13 93

2 .4 6 0 a’ $ c o s x 1 3

31 . Si n x 2 V— 1

‘ / T j . e s e c o s x 0 2

F o rmu lae o r erentia ti o n o Inverse Tri o no metri c 1 9 . f Diff f g

F uncti o ns .

” 1 sin u

—l XXI . co s u

‘ l u XXII . t an

“ l I I u XX I . c ot

I ‘ 1 XX V . se c u DIFFERENTIA TION 33

‘ 1 XXV. c o seC u

* 1 XXVI . vers u

eri va ti o n o Fo rmulae . 20 . D f

‘ l Let : n u P ro of of XX. g si

n therefore si g u .

g da B I COS ﬁ y XII , g dx dx

du therefore dx c o s g

alu

“; therefore dg d 2 d“? u

T ma b e i k P ro of of XXI . his y derived l e the relation —l “ c o s u s 1n u

d —l d - 1 whenc e co s u s m u dx dx

‘ l II Let z : an u P ro of of XX . g t ;

therefo re tan g u .

2 dg du B s e c y XV ’ y ’ dx alx

dg dx ° therefore 2 dx sec y 34 DI ERE TIA L A L FF N C C UL US.

2 = 2 z sec y 1 + tan g = l + u ; du

dg dx therefore 2 dx 1 u

P ro o o T m f f XXIII . his ay b e derived like the relation “ l “ l c u an u ot t .

—l P ro o o X Let s e f f X IV. g c n ;

s ec u therefore g .

d da g __ B X . e an y VII , s c g t g dx dx

dg dx therefore dx s e c g tan g

2 s ec g tan g s e c g se c y 1 u

du d therefore ?! g“3 f u Vu 1

P ro o o X T ma be X k . f f XV. his y derived li e XIV , the relation ‘ l ‘ 1 c o se c u s e c u .

“ 1 Let = v e rs u P ro of of XXVI . g ;

r u 1 c o s the efore vers g g .

alu B s m y XIV . y d,

du therefore dg dx dx Si n g

z 2 But c o s gy u ) du

dx therefore I ERENTIATI 35 D FF ON.

EX A M P L ES.

' l Z = - : tan mx. 1 . g Z

“ fi 3 = sin 3 x 2 . g (

2 ‘ 3 s . g ver

‘ 1 — 3 : sin 3 x 4 x . 4 . g ( )

tan 5 . g

-l ' tan o . 6 . g

" an n tan x . 7 . g t ( ) m

co c 8 . g se

’ -1 2 x 10 3] VOTE o 1 + x2

e’ —6 ' l 11 y = tan 2 36 DIFFEREN TIA L CAL C UL US.

l — 2

u x " 4 si y = tan 3 + 5 co s x

3 5 c o s x ' l y co s . 5 + 3 co s x

2 1 _ x ‘ 1 = s in g i l + x

-- 2 1 l I x g c ose(3 2 x

1 —d x

“ = an 22 . y t

38 DI ERENTIAL CAL C UL US FF .

Fo r xam u e ple, s ppose

a x g 1

Diﬁ e rentiatin c g with respe t to g,

d x a

a By (1 by 2 dz a

This is the s ame result that we get by solving (2) with nc to n refere e g, givi g a y —l E r an f n a n n d di fere ti ti g this with refere c e to x .

i ! ig dz To ex ress ii i n terms o g and a n 2 2 . p f If g is give dx dz dx

unc n o f z and z a n un c n o f x a f tio , give f tio , it follows th t

un n n n g is a f ctio o f x. This relatio may be obtai ed by y m na n z b n n ua n but c an be eli i ti g etwee the two give eq tio s, gx n n found withou t su ch elimi atio . 1” B n a n tw o n ua n ﬁnd 94 and y differe ti ti g the give eq tio s, we dz 1 g £3 an d m ff n al c c n d_ma be b a n , fro these di ere ti oefﬁ ie ts, y o t i ed dx dx by a relation whic h may b e derived as follows

Ag Ag Az n a It is evide t th t ’ Ax A z Ax

A A ma Ax A and z. s u an a ac however s ll , g, these q tities ppro h z a fo r m o f m mb o f u a n ero, we h ve the li its the e ers this eq tio ,

1 dx dz alx

T a a n am as di n a co efﬁ h t is, the rel tio is the s e if the ffere ti l n cients were ordinary fractio s . DH ERE THMU . M N ON .

Fo r xam u e ple, s ppose

s e , 2 2 2 ( ) 3 a x .

f n ia n ua n nc z Dif ere t ti g these eq tio s, the ﬁrst with refere e to , an d c n n c x a the se o d with refere e to , we h ve

dz dx

4 — — 2 — 2 4 By 5 z 2 x = l 0 x a x b 31 ( ) ( ) , y

Th e s ame result might have been Obtained by eliminating z between giving 2 a ( 06

n n n n x a d differe tiati g this with refere c e to .

EX A M P L ES.

In the following seven examples ﬁnd by differentiation 2 A 21 . and then by (1 ) rt . 3£13 ““L 2 a I dy (y dx 2

? dy V a 1

2 2 x —1 = 2 1 au x co t x ) EV ?! (t ). 2 DI ERE TIA A 40 FF N L C LC UL Us .

x 2 lo 7 . g

In the followin g e x amples ﬁnd by differentiation 0131 1 A rt 2 2 . and then by ) . dz: (

2 z dg 4 — — ' 3 5 2 2 x 1 d x ( x d = — 2 y 3 2 l o x x . 9 . z g ( ) 4 x 6 x 1 . dx

22 d 5 e — 3 3z g = lo z% — z z = e 10 . g g ( ) , 2" d x e —1

2 "° — ‘ "c 1 + z d e e = = " g l z e . 11 . o , g g ’° ' ac z atx e - e

2 d 2 x — 2 x + 1 " 1 g : = 12 . an 2 z z tan 2 x g t , ( — 2 2 dx 2 (x x ) CHAPTER IV .

S U C CES S IV E DIFF ER ENTIA TION .

m 2 e niti o n . A S n n a n o n 3 . D ﬁ i gle differe ti tio perfor ed

7 = x f n a c c n 1 T s u g f ( ) gives the dif ere ti l oeffi ie t, his re lt %x b n n a a a un c n o f x m a be a a n n ei g ge er lly lso f tio , y g i differe tiate d an d u b a n a c a seco n d erentia l , we th s o t i wh t is lled the d ifi co efficient; the result o f three suc c essive differentiation s is

' thir d erentia l co e i ient n the d iﬁ ﬁ c a d so o n .

4 Fo r xam x e ple, if g ,

dy 4 x3 , dx

d 3 2 4 12 S , dx dx

d d d t/ 4 £23 2 . alx dx dx

h No ta tio n. T e c n n 24 . se o d differe tial coefﬁcient o f g with y c x n b — respe t to , is de oted y z gx

T a h t is, 2 dx dx dx

d2 S m d g i ilarly, 3 2 dx dx dx dx dx dx

4 d g d d d dg d ﬂ . 4 3 dx dx dx dx dx dx dx

n n ‘ l d g d d g ‘

" ‘ 1 dx dx alx" 42 DI ERE TIA A UL US FF N L C L C .

T u h s, if

dx3

Th e successive differential c oefﬁcients are sometimes called hir irst s eco n t eriva tives . the f , d, d, d n a unc n o f x n b x suc If the origi l f tio is de oted y f ( ) , its c essive differential c oefﬁcients are Often denoted by

x w x f ( ) , f ( ) , f ( )

’ b x The nth D erenti al Co e ici ent. 25 . iﬁ ﬁ It is possi le to e press

the nth differential c oefﬁcient o f some functions .

Fo r xam e ple,

F m = ’ a a . e ( ) ro g , we h ve

“ m w e a b . F c ( ) ro g , h ve

i d v 2 alx

F m lo x a ro g g , we h ve

2 d3 1 - 3 2 3 35 (ti90

"" 1 n —1 n —1 id g d g L_ _ 4 “ dx dx SUCCESSIVE DIFFEREN TIA TION 43

F m sin ax a ro g , we h ve

di ] a co s a x = a sm d x + dx

2 d il 2 2 a co s a Si n ax 2 dx

B d y — 3 3 a co s a Si n x + 3 dx

EX A M P L ES .

4 3 2 — 2 = — — 2 x . x 1 1 g x 4 x + 6 x 4 + l . ZZ (

= 5 x . 2 . g

’ 3 . 4 g xe + x.

= 5 . g x lo g x.

= 3 . x 1 6 g 0 g x.

3 7 . g l o 6 + g ( ’ 3 (e e " ) 44 DI ERE TIAL AL UL FF N C C US.

i — 1 ” 8 . y sc 6 90 + 2 e c w

i d y o x x lo x . g ” g dx

3 = _d_ y 2 c o s x lo s 1n x . 10 . / 3 g 3 3 dx siu x

3 3 2 53 d g 4 a 11 . x a tan y s dx

d ‘ g - " 12 . 4 e co s x . a};

= ﬂ = ‘ — ’ tan x . s x 4 x 13 . g ﬁ e c s e c . dx3

° 5 x + 1 d g 3 . = § 2 — 6 L x 1 dx (x + 1)

Dec ompose the frac tion b efore differentiating .

z d g “ 1 s e c 2 w 5 . y v a a cm

ig 2 g y iz 16 — n —4 n n -1 T g 2 g ( ) g dx

8 3 7 c s x c s x (l 3 Siu x g qz7 cing

d z = 2 ’ g g tan x + 8 10 g c o s x + 3 x . Egg

2 g = (x dx8

d s 3 2 ?l 2 g = x 3 (lo g x) -l l lo g x +

d 2 d as y — y 2 2 = e Si n bx 2 a + a + 0 0 . g 2 ( ) g dx dw

2 ( — 2 g -- ’ ( 1 x ) x ggI m g : 2a

46 DI ERE TIA A U S FF N L C LC L U .

Differentiating (2)

d2 uv = u v u v + u v uv = u v 2 u v z( ) 2 1 1 l 1 + 2 2 l 1 + dx

3 uv u v u v + 2 u v + 2 u v u v + uv 3 ( ) 3 2 1 2 1 l 2 + 1 2 3 5x

U ?) 3 u w 3 2 1 + 3 “U3 .

a ﬁn d law o f m a far We sh ll this the ter s to pply, however c n nu n a n c c n n o f we o ti e the differe ti tio , the oefﬁ ie ts bei g those B n m a T m the i o i l heore . In general — n (u l ) uv “” nu 21 n v ( ) n n 1 1 + u —2 2 L2_

v u v w 3 l a - 1 + e. ( )

T ma be b n uc n b n a u his y proved y i d tio , y showi g th t if tr e for t d n + 1 uv a u T x c ( ) , it is lso tr e for his e er ise is left for f$ 71 dx71 n the stu de t . In the ordinary n otation (3 ) be comes

d” n

“ l " ‘ 2 2 dx" dx” dx" dx [2 dx dx

‘ 1 ” du al" v d v

“ 1 " dx dx" dx

’ Fo r xam u s ﬁnd b L bn z T m e ple, let y ei it s heore

“ “ u : e u = a c Here , 1 ,

= x v = 1 v = 0 v = 0 v , 1 , 2 , 3 ,

Substituting i n we have

d n ‘n ” “‘ " “ 1 “ " “ 1 “ (xe ) a e x na e a e (ax n ) . dx" TIA TI S UCCESSIVE DIFFEREN ON . 47

EXA M P L ES .

’ Fin d by Leibnitz s Theorem the following diﬁ e rential co efﬁc ients :

3 3 z 2 z z : x tan w 2 x $ ec x 3 tan x 1 x s ec x tan a 1 . y . ( )

z w t n w 18 x sec 6 a .

4 6 8 ’ ”c m w. e c 2 . e o y g g 5

" 2 l x l a ( o g [( o g a n ) n].

n 3 ( a; l ) + APTE CH R V .

DIFF ER EN TIA L-S

13 2 7 The f n a c c n ! h as b n n no t as a . di fere ti l oefﬁ ie t ee deﬁ ed, 27a: ac n h av 1n a num a and n m na but as a sm l e fr tio g er tor de o i tor, g

symbol representin g the limiting valu e o f as Am and Ag Ax a ac z B ut are m a an a in a n ppro h ero . there so e dv t ges reg rdi g

n a “ c c n as an ac ua ac n doc an d cl the differe ti l oefﬁ ie t t l fr tio , y

' b n n n ma n c m n o f a: an d an d c a a er ei g i ﬁ itely s ll i re e ts y, lled iy

n f n T a da: an i n n itel s ma ll Am and . e ti ls o a: a d . a y h t is, is ﬁ y , dy an i nﬁni tely sma ll Ay. 2 F r n n w e f n a : 90 b a n o i sta ce, if di fere ti te y , we o t i

dy 2 x d x

n n a u m be n U si g differe ti ls, this res lt ight writte

m dy 2 a cl .

n m n A These are two forms o f e xpressi g the s a e relatio . c c n ordi g to the ﬁrst,

h i o the i ncrement o to that o so a s The limit of t e ra t o f f y f ,

s a r a ch zero i s 2 90 . these i ncrement pp o ,

n c n A cc ordi g to the se o d,

A n inﬁni tely sma ll i ncrement of y is 2 90 times the co rresp o nding inﬁn itely sma ll i ncrement of a .

We have the s ame two forms of expressing other relations

in mathematic s .

r i n an w e ma s a Fo st c e, y y

The limit o f the rati o as these quantities approach ” z un . ero, is ity DIFFERENTIALS. 49

r O ,

m r u h An in ﬁnitely s all a c is eq al to its c ord .

Th e equ ation cly 2 a an may thus b e u sed as a convenient sub stitute for dy

clx

' f a wh o r c a th e d erenti a l co e ici ent We see lso y is lled iﬁ ﬁ , w

‘n o f dcc i n u n d w fo rit is the co eﬁ cie t the eq atio y 2 dw.

Th e mu a fo r f n a n ma be x in 28 . for l e di fere ti tio y e pressed

o m o f r n a b m n doc in ac m m the f r diffe e ti ls y o itti g the e h e ber .

I b c m T u V. h s, e o es d (uv) 71t a dv ;

—l da an d I tan i t XXI , d 2 ; 1 711

n m m a x a d the o thers ay b e si il rly e pressed . Differentiation by the n e w formulae is substantially the am as b o ld f n n in u n m s e y the , di feri g o ly si g the sy bol d i instead o f doc

F x m k x o r a 1 . a E . 5 . 7 e ple, t e , p

dy = d (m u s )?

2 ? ( 90 + 3 ) (m 3 )

‘ Dividin g by also gives

— — 2 dg/ 3 6 50 93 n2 da: ( . 50 ERENTIAL A L DIFF C C UL Us .

ff n Successi ve D erentia ls . Succ a co effi 29 . ifi essive di ere ti l 2 3 d y d y c 1 n m i c a b n n as Si n mb e ts, g, , wh h h ve ee defi ed gle sy ols, bl? {fi } 2 3 ma a b e n as ac n num a cl d y lso i terpreted fr tio s, the er tors, y, n n cl cl and c a c n d m , de oti g ( y) , m , lled the se o ,

o n n a o f n o m na o are da third, , differe ti ls y, While the de i t rs ( y,

T b e b un m an x m his will etter derstood fro e a ple .

4 Let (13 y ,

3 n d 4 x w the y d .

3 A s 4 x dx a a ab cl a a ab and ma b e is v ri le, y is v ri le, y n n No w a: n n n a a a . b n a ab g i differe ti ted , ei g the i depe de t v ri le, its increment elm m ay b e su pposed the same in ﬁnitely small u an all a u o f a a ma a dcv as q tity for v l es ; th t is, we y reg rd n an in n n T u b a n c o st t the prec edi g equatio . h s we o t i

? d (dy) 12 a do: dx 12

2 Deno ting d (dy) by d

z d y 12

ff n a n a a n and a n Ola: as c n an Di ere ti ti g g i , still reg rdi g o st t,

z 2 d (a y) 24 xclx (da:) 24

3 cl y

F m u a n b n b da: in ro these eq tio s , y dividi g y the power of the c n m mb ﬁnd se o d e ers, we

? d 2 12 90 2 , (cl /$ 3 d y 24 x. (an) 3

The n n n a ab 93 f n a u i depe de t v ri le , whose di fere ti l is s pposed n i m c n a s o m c a e ui crescent a ab . o st t , s eti es lled the q v ri le DIFFEREN TIALS. 51

EX A M P L ES.

Diﬁ erentiate n u n n a s in c the followi g, si g differe ti l the pro ess

2 90 + 2

93 1

3 .

x " l e l o x. ol e o x Ola) y g y g .

' “B ‘ —e ez — e

“ e" + e ’3

‘ m n m l ' 2 z : sin s x in a a' i x co . s c n s n x el y os ) m.

1 mm t n 4 a ac. d e x lx. 35 y s c c

dx ‘ 1 8 = tan 1° w y g 2 wt1 + 1 C AP E H T R VI.

IM PLIC IT F N TI N . ee als o A rt U C O S ( S .

2 3 in n n h b n n x 30 . o as a c Hitherto ﬁ di g s, , y ee e pli it (dig

n a n b n n fun ction o f (17. Whe the rel tio etwee y a d a: is given by an equation c ontaining these qu antities but no t solved with li n nc a b e an im ci t u c n m. refere e to y, y is s id to p f tio of

u a n c an be o r n c If the eq tio s lved with refe e e to y, we may ﬁnd its differen tial c oefﬁcients by the methods already

n But u n no t n c a fo r f n a n g ive . this sol tio is e ess ry the di fere ti tio , for by the u se of the formu lae of differentiation we may derive 3 il m c m n ua n . , 3, , dire tly fro the give eq tio ga 35

1 Fo r xam u a n b n and a: to 3 . e ple , s ppose the rel tio etwee y b e given by the equation

2 2 2 2 = 2 2 a y + b x a b .

f n a n c x Di fere ti ti g with respe t to ,

d 2 2 2 2 a 6 93 O ( y + , iia:

2 ﬂ z 2 b n O 2 a y , ga:

dy a : g doc a y

n a c c n Having thu s obtain ed the first differe ti l oeffi ie t, we ma b ff n a n a a n c o n f n a y, y di ere ti ti g g i , derive the se d di fere ti l n c oeffic ie t . d 2 2 _ 2 2 z/ a yb 6 a:(1 da:

4 2 a y

54 N CAL C UL S DIFFERE TIAL U .

d c o s w __y g + y) dx 1

da . +y _. 6 m ‘ y ? 2 da: x (y ax x (y

z n 2 % = dy ta a; d y tan y tan w m . 7 . s e c c o s y 2 3 Ola: tan y (190 tan y

2 z 3 2 a 3 s dy x a n d y a y 8 x 3 a n O. , y y f ? 2 3 doc y am am (y a n)

a b co s 0 a 0 b n 0 , y si ,

a ab b n a an d 0. the v ri les ei g , y,

z dy a + b c o s 9 a y ? Ola: b sin O ax CHAPTER VII.

EX P A NS IO N O P F U NC TIONS .

‘ 2 The u n bab a a am a m 3 . st de t is pro ly lre dy f ili r with ethods T u b n x n n n n n . a o f e pa di g c ertain fu c tio s i to series h s, y ordi ry n divisio ,

3 1 — a' +

b B n m a T m y the i o i l heore ,

— " ” ‘ 1 M n 2 2 (a as) na a: a 93 +

But these methods are limited in their application to c ertain

r n o w b u c n m forms o f function s . We a e a o t to o sider a ethod o f x an n a cab all unc n an d nc u n as e p sio ppli le to f tio s , i l di g x spe cial c ases the e pan sion s j u st referred to . ’ These methods are kn own as Taylo r s Theo rem and Ma c~ ’ la urin s Theo rem T o m are c nn c a . hese two the re s so o e ted th t ma e n n a either y b regarded as i volvi g the other . We sh ll ﬁrst ’ c o nsider Macl au rin s Theorem as the simpler in expression nd n a derivatio .

’ M l urin ac a s Th . T a m n 3 3 . eo rem his is theore by which a y function o f a: may be expanded in to a series of terms arranged acc n a c n n n a ordi g to the s e di g i tegral powers of . It may b e expressed as follows

f (x) =f (0) “m m + I2 I3 in c a n un c n be x an an d x whi h f ( ) is the give f tio to e p ded, f ( ) , n u cc r n a c c n f ( ) , f m , its s essive diffe e ti l oefﬁ ie ts . 56 CA L C UL US DIFFERENTIAL .

’ T a x i o h t is, f ( ) f c ) cla:

i ' ﬂ ee) f a) dx

f (n— gi

O as n a n m n f ( ) , m , the ot tio i plies, de ote the

a u a n v l es ( ) , m , whe

’ eri ti h T m a e 4 va o n o M a cla rin s T eo rem. b 3 . D f a his y de rived by the metho d o f Indeterminate Coefficients by assuming

2 4 Cx + Dx3 + Ex + (1 )

' A B r f n C a e u b e co nsta nt c c . where , , , s pposed to oe ﬁ ie ts n a n u cc and u n n a n u Differe ti ti g s essively, si g the ot tio j st de n w e a ﬁ ed, h ve

’ 2 3 f (w) B + 2 0 90 3 Dx + 4 Ex +

" - 2 f (w) 2 3 Dx 3 . 4 Ex +

2 3 1? 2 o 3 -4 Em

No w sin c e equation an d c on sequ ently

u ru fo r all a u o f a be u n s pposed t e v l es , they will tr e whe

Sub u n z ac in ua n a stit ti g ero for these eq tio s, we h ve

from or

o r

’ —Ji l Q) o r C ’ Z t- X OF E PANSION FUNCTIONS . 57

from o r

iV O = 2 -3 -4E 0 1‘ f ( ) ,

Sub u n a u o f A B C i n a stit ti g these v l es , , , we h ve f (w) =f (0 ) f

’ A s an xam in a ca n o f Maclaurin s T 3 5 . e ple the ppli tio heo

rem b e u x an lo 1 a n a . , let it req ired to e p d g ( ) i to series

1 — f a) (1 + x) f (0 ) 1 ~ — 1 ~

iv =— r L3_.

f a) s o oars f " (0 ) I 4.

i n i Sub u n 6 Art . 34 a st t ti g ( ) , we h ve

2 3 x 2 90 -a:

4 5 96 o r 4- 2 5

’ i n a ca n o f Maclaurin s T m a n 3 6 . If, the ppli tio heore to give ' unc n an o f uan O O O are f tio , y the q tities f ( ) f ( ) f ( ) ,

n n i unc n no t a ab b n x an i ﬁ ite , th s f tio is c p le of ei g e p ded m the

' d . T IS a l o 03 a a c n c . propose series his the se with g , , ot DIF E ENTIA L CAL C’UL F R US.

E A M P L X ES .

’ Derive the following by Macl aurin s Theorem

3 5 7 = cv x x 1 . s 1n w w

I_3_

= M l o e . 5 . , where g a

3 5 $ 2 93 6 t an w cc + + 3 15

3 5 7 m 5 ‘ l = — 7 . tan x w + % 5 ;

- l a tan oo f( ) ,

1 — ‘2 4 e m 1 —a -—w —m -- m f ( ) i I , 1 + w2

" 3 5 f (a ) 2 a: + 4 x 6 90 +

s 6 i 90 S: 8 . I 3 5 2 -4 -6 7 X OF E PANSION FUNCTIONS. 59

-1 90 sin a f ( ) ,

1 I f w (1 W

Ex andi n b B n m a T m p g y the i o i l heore ,

2 1 4 1 + x + 90 2 2 2 6 212

z 4 6 1 an + bn + ca +

1 3 a = i e where , ’ ’ 2 QTZ 2 -4 -6

3 5 v f (90) 2 a m 4 bn + 6 c. +

4 6 $ - 1O co s a: " 4 42 4 gm I where M 3 9 5. 12 45

m 12 . F Ex ro . 7 derive

1 1 1 1 + 3 5 7 9

—1 —1 1 -1 1 Al nc tan 1 _ tan 3 tan so, si e 2

21 50 3 7 6 . 785398 60 DIFFERENTIA L A C L C UL US.

The c ompu tation includes 10 terms the ﬁrst series o f o n 7 the se c d .

m x F E . 3 13 . ro show that

x~ w r l — e l +

c o s x 1 sin x b E xs 1 2 . , y . ,

S m a a i il rly , show th t

c o s w i n 1 s w.

From thes e two equ ation s derive the exp o nenti al va lues o f n an d c n the si e osi e,

s in x _

— z ﬁ i —l i e + e COS CB 2

’ Ta lo r s Theo rem. T a m fo r x n n 3 7 . y his is theore e pa di g any f unction o f the s um of two quantities i n a series arranged n o n e o f uan acc ordi g to the powers of these q tities . A s the Binomial Theorem e xpands (ao in a series ’ a an ac c n h Ta T m rr ged ordi g to the powers of , so ylor s heore x an a n unctio n o f 93 h in a m a ma e p ds y f ( + ) si il r series . It y be expressed as follows

(e ) + f

’ Th e Ta T m n u n 3 8 . proof of ylor s heore depe ds po the fol lowing princ iple

If f r n a ac h nc w a n h we di fe e ti te f ( ) with refere e to , reg rdi g c n an u am as f n a o st t, the res lt is the s e if we di fere ti te it with

nc h a n a: c n an . refere e to , reg rdi g o st t

62 A L DIFFERENTIAL O OUL US.

Equating the c oefficients o f like powers o f h accordi ng to the nc o f n m n a Co c n a pri iple I deter i te efﬁ ie ts, we h ve

The c oefﬁcient A may b e foun d from (1) by putting h as the equ ation must hold for thi s value among others .

Then A f

(1A H enc e f (x) dx

1 d2A 1 93 2 f ( ) 2 de 2

' n x n A B C in w e Substituti g these e pressio s for , , , have

’ ’ Macl aurin s T m ma be b a n m Ta 40 . heore y o t i ed fro ylor s a: n a Theorem by substitu ting 0 . We the h ve

h? E

’ ' ‘ This is Maclaurin s Theorem expressed in terms o f h in s tead o f x. EX OF PANSION FUNCTIONS. 63

A s an xam in a ca n o f T m 41 . e ple the ppli tio heore , let i t be required to expand sin (a: h) into

a: h n n: h f ( ) si ( ) ,

f (a?)

' w co s x f ( ) , f f f i" (w)

n i n ﬁn d n x 2 Art . 3 9 Substituti g these e pressio s ( ) , we 2 } 3 4 —i -f— — s in ( a: -h) = sin czz-t-h co s cc c o s x + sin w+ m [2 l§ %

EXA M P L ES .

’ Deri ve the following by Taylor s Theorem

h — h2 h3 h4 1 . 2 3 2 w 3 x

n j ‘ l n " x" nwn h ﬁ — — ) se 2 . ( a: h) é

[Q s IL — —E --— - m 3 . h s in x c o s w f sin x F . 2 L_ 3

2 2 2 tan a h tan x h x tan x 4 . ( ) sec a: h s e 0 3 1 z' z ; se c cc (1 3 tan x) h2 h3 “h z 5 . e e 1 h + + l2 l§

2 l o in 6 . g s l o g sin x + h c o t a: co sec w+ DIFFEREN TIA L CA L C’ UL S U .

2 l z 7 . o g sec (a: h) l o g s e c cc h tan x gs e c x

3 4 l g g -é-s e chvtan a: {é s e c wﬂ 3 tan w)

’ ’ 2 The c n o f Ta and Maclaurin 4 . pre edi g proofs ylor s s Theorems by the method o f Indetermin ate Coeffic ients are n o t a r a ac o r n a mu c as o b o f o ltogethe s tisf t y, i s h the p ssi ility devel p m n in o m um e t the prop sed for is ass ed . ’ A n u o f Ta r T o m n n n o f y rigoro s proof ylo s he re , i depe de t n m n a f n m C c c o a a cu . I deter i te oe fi ie ts, is p r tively diffi lt We give n as n n a f u u n the followi g prese ti g the le st di fic lties to the st de t .

Co nti nuo us F un t un o n a be c o n c i ns . A c 43 . o f ti is s id to tinu o u s b n c a n u the n n n a ab etwee ert i val es of i depe de t v ri le, when it chan ges gra du a lly while the variable passes fro m o ne

a u In o a co n ti nuo us unc n v l e to the other . other w rds, f tio is o ne a n n u th t c a be represe ted by a co ntinuo s curve .

44 a n unc n ac z n a: a and n . If give f tio (M ) is ero whe whe at : b and n and o ntinu Ous b n a u as , is ﬁ ite c etwee those v l es , well as its differential c o ' efﬁcient then qS (a ) mu st b e zero for s o me n value o f as between a a d b. Let the function b e rep resented by the c urve L A ==a et G , == T n r n 0 B b. he acc o di g

X to o 0 the hyp thesis , y

n a) : a an d n a : b. whe , whe n n nu u b n A an d B mu Si ce the curve is co ti o s etwee , there st m n n m an n a a be so e poi t P betwee the , where the t ge t is p r llel : nc K an d c n u n 0 . See Art to O , o seq e tly ( He e the proposition is established . EX PANSION OF FUNCTIONS. 65

’ aid o f ro o siti o n Ta l o r s T m can no w With the this p p ‘ y heore u n m n ﬁ n be derived witho u t the se of I deter i ate Coef cie ts .

’ h x and P ro o o Ta lo r s T eo rem. Su su c 4 5 . f f y ppose f ( ) its c e s s iv e n 1 differ ential c o efﬁc ients to b e ﬁnite and c ontinu ou s

: n Le between cc a a d a: a h. t

wher e

— " — a + h a a m a > . f < > ( r Em )

b e n o c a n n n It is to ti ed th t R is i depe de t o f a .

It is evident that when w = 0 an d when

nc b A rt . 44 o m a u o f a: n He e y , for s e v l e betwee 0 h’ n . o a d h Supp se this valu e . Then — wf ( a )

' “ R : 0 n a : h n , whe .

B ut 0 when cc : 0 ; hen c e qS (ac) O for some valu e ' as b n n of etwee 0 a d h . Contin uing this proc ess to n 1 differentiations we ﬁnd

w) R o fo r m a u at b n 0 n L so e v l e of etwee a d h . e t this value o f a:

b e 0h 9 1 . , where

l T n ” + he f (a 6b. ) R .

E ua n a u R a n ab a q ti g this v l e of with th t give ove, we h ve

' < a >

1 ” + a h f ( 9 ) . 66 DIFFERENTIAL CALCUL US.

ma n o w ub u a: a nc a ma a n We y s stit te for , si e y h ve a y an d we have

n -H h 1 ” + x h + f ( 0 ) . ln + 1

’ emai n er i n 46 . R Ta lo r s Th Th d y eo rem. e last term

u + l 1 W “ 9h) | ri+ 1

ca ma n a m n m o f is lled the re i der fter n 1 ter s . Whe the for unc n u uc a b ak n n u c n a the f tio fl ) is s h th t y t i g s ffi ie tly l rge, ’ ma n can b e ma n n ma n Ta this re i der de i defi itely s ll, the ylor s n Theorem gives a c onverge t series .

’ F a ilure o Ta lo r s Theo rem. n cc o r an 4 7 . f y Whe f ( ) y of its suc c essive differential c oefﬁcients ar e inﬁnite or dis c ontinuo u s

n n d ax h c n mo n a o n no n b etwee a: a + , the pre edi g de str ti lo ger ’ and fo r uc a unc n Ta T m a holds good, s h f tio ylor s heore is s id to fail .

’ a: O in emain er i n Macla ur in s Theo rem. 48 . R d If we let

n u a n a the precedi g eq tio , we h ve

n a; h Or substituti g for ,

G f ee) =f (0 ) ar e ) EN )

k u n When the remain der f by ta ing n s fﬁcie tly

a c an be ma n n ma c o n g n . l rge, de i deﬁ itely s ll, the series is ver e t OF EX PANSION FUNCTIONS. 67

R ema i n er i n certa i n s eri es . Le t u s a n a 4 9 . d pply the ge er l 1 a; 71+ x r o n fo r ma n l e p essi the re i der, to the deve op f ” men t o e . Here

Th e fraction can b e made as small as we please by

ak n n u c n a a r ma b e a u o f 93. t i g s fﬁ ie tly l rge, wh teve y the v l e ” M r e n nc R a r ac z . o eover, is ﬁ ite ; he e pp o hes ero Henc e the series 2 x x3 ez = 1 + x + -— [2 IE ll is c onvergent for a valu es o f x .

n ca “ " + 1 i s i n a 090 i a z r i ts i mi It ev de t th t f ( ) W ll h ve e o for l t, 1 when ever fl u) is o f su ch a form th at all o f its s uccessive dif f r n i l c c n are n T c a n ce and e e t a oeffi ie ts fi ite . his is the se with si nc x an n co s a . He e these e p sio s

3 5 x sin w = x —£ +

2 4 x x c w x = 1

n ll f are c onverge t for a valu es o a . x l o 1 x n ma n If f ( ) g ( ) , the the re i der

" f “ — 1 ﬁ o ) l | n + 1 (1

This may be expressed as

R u + 1

If a: an d ua o r an u n R has a is positive eq l to, less th , ity, limit o f zero . 68 DIFFERENTIAL CALCUL US.

Hen c e the e xpan sio n

4 2

c n n fo r a u o f a n x 1 o r x 1 but is o verge t positive v l es , whe < ,

n n w 1 . diverge t, whe

DIFFE ENTIAL A R C L C UL US.

2 50 ” f 2 V “L Fo r n anc c n ac n i st e, o sider the fr tio i

W n a: 2 ac n ak m he , the fr tio t es the for 3

1 ac n ak m : 0 , the fr tio t es the for 3 0 .

a: 1 ac n ak m w c indeter , the fr tio t es the for 8, hi h is

TO eva luate a ra ctio n tha t takes the i n 9 5 2 . f determi nate f o rm

Frequently an algebraic transformation in the given fraction m n a u will deter i e the v l e . If the fraction i n the prec eding a c be uc m a u c w as b rti le red ed to lower ter s , its v l e, whi h efore 1 n m na n a; 1 b e un to b e . i deter i te whe , will fo d

513 —2 A s an u a n c n ac n other ill str tio , o sider the fr tio Va: 1 1 0

W n 93 2 ak m . B ut b a na z n the he , this t es the for 6 y r tio li i g

n m na an m ac n n de o i tor, we tr sfor the fr tio i to

a: 2

c m 2 n x 2 . which be o es , whe

The f n a Ca cu u u n n m 53 . Di fere ti l l l s f r ishes the followi g ethod

applic able to all cases .

i o r the nu era to r a nd eno mi na to r res ectivel Su bst tute f m d , p y,

' i co e icients The va lue O thi s n ew ra cti on o r their diﬁerent a l ﬁ . f f f

the a ss i ne va lue O x will be the va lue re ui re . g d , f q d

n n to : a To u ac i , whe ; prove this, s ppose the fr t o 33g; 8

a a 0 and 0 . th t is , ¢ ( ) ,

f n m o f B Art . 50 u a u o ac y , the req ired v l e the fr tio is the li it (Na -H t ) as h approaches zero . ﬂ a w-h) M INDETERMINA TE FOR S. 71

’ B Ta s T m y ylor heore , h2 h3 h + (9 0 (” C 0 0 0 m E W M 1 W ) r h r ( 90 ) + M o (2 ll

ub u n a x and m mb n a S stit ti g for , re e eri g th t we have h2 I J}; m ( a ) “ re W ” ) h? we ) ” ( a ) ( a ) _ | g

as h a o ac z therefore, ppr hes ero,

lIf ( a h) W(a )

a O and 0 a m a m If ( ) , , we h ve si il rly fro as h a ac z ppro hes ero,

- Ma i h) ll! ( 06)

a c mu b e a an d as n as ma th t is, the pro ess st repe ted, ofte y b e n ec ess ary to o btain a result which is no t

Fo r xam u s ﬁnd a u o f r ac n in A rt 51 e ple, let the v l e the f tio . ,

2 cc) x 4 3 x + 2 9 n a: 1 ’ whe . 519 — 1 0

( WW) 2 a: 3 1 nc n a: 1 . He e ’ whe _ 2 a: 2

Fo r ano xam u s ﬁnd a u o f ther e ple, let the v l e

x - x ( x —2 f ( ) e + e 9 m= n O. , whe t]; ( a ) 1 c o s a:

c ” ; O n a: O ’ whe . s in x O

N ~ ¢ < x) ex + p r 2 n a: O , whe . ’ ‘ a we ) u mft, 72 I N I CAL L D FFERE T AL C U US.

EX A M P L ES.

F ind the v alues fo llo wing frac tio ns

lo g a: : 3 n w 1 . n whe A s . 1 . 513 — 1

£17 — 2 w n a: 2 ’ he . ( a: 1

‘” ‘ “B e e A s 7 n . 2 . s in a:

A ns .

t a n (x 1)

A ns .

to : ( 7r 2 m)

A ns .

4 3 90 4- 2 93 4- 2 50 —1 w = 1 6 2 ’ w — 15 w + 24 x —1O

2 tan .v s in 2 w x = 0 3 s in a;

L ‘ 5 ” zz +3 x ’ 1 5 C 10 e 15 e 6 6

2 x + 2 4 6 6 3 8

z s ec cc 2 tan a:

12 . l + c o s 4 x

“ A ns . 6 e . “ z ’ (a: 4 ) e e x INDETERMINA TE FORMS . 7 3

93 3 f A ac n ma ak o f m . 54 . fr tio y t e either the for s, , , a 0 0 00

B a n a u o f a ac n as a m n y reg rdi g the v l e fr tio li it, it is evide t 3 t a in c a 0 0 and h t the ﬁrst two ses , , 3 00

Th e m n m na a n a for 22is i deter i te, for the re so th t, if the numerator an d denominator both increase beyond any ﬁnite m a n n o t u c n m n m t o f li it, this lo e is s fﬁ ie t to deter i e the li i the n fractio .

To e va luate a ra cti o n tha t ta kes the o rm 55 . f f

Suppose when a : a ; 41 013) = a a co and a co . th t is, ¢ ( ) , d( )

By taking the rec ipro cals o f (Mr ) and 1

n m= a $52; whe .

nc b Art . 53 He e y ,

di sc) m n a u o f n a: a a u o f the li iti g v l e , whe , is the v l e if ( 93)

(M90) n z whe m a . WW) 93) div (cc)

lbw) T a h t is, 511 W) ( a )

' a) n( a ) we ) e ra ) i . nc l z , he e , Ma ) WW) 74 DIFFERENTIAL CALCUL US.

In n a 1 b derivi g we h ve divided ( ) y If, however, a

o r 0 0 ua n 2 n o t c a fello w m , eq tio ( ) does logi lly fro

N ma be o n a 2 u in evertheless , it y sh w th t ( ) is tr e these c a a ses lso .

Su M O an d n a n u an ppose , ﬁ ite q tity, l11 096)

Ma ) Ma )

To a ac n 2 n a this l st fr tio , ( ) evide tly pplies, therefore a Mar) ii! ( )

' m M a ) we ) e ga ) e H , M o r , ' Ma ) llf ( a ) W(a )

(Na ) 00 n M 0 , the , Ma ) Ma )

n ca an d we have the precedi g se .

Thus the fo rm g i s eva luated i n the same way a s the f o rm

F r xam ﬁnd a u o f o e ple, the v l e l o g x when a: O. c o t a:

> x l o x oo 4 ( ) g x O when . c o t a 0 0 r 1

2 a: sin a3 0 n a: 0 ’ whe ° Q ' 11/ (cc) c o sec a x 0

( M r ) 2 sin wco s x 0 — g when x O r" (w) 1 1

-0 TO eva lua te a uncti o n tha t ta kes the o rm 0 0 . 56 . f f

m na n o ne Th e p ro du ct ¢ (w) bec omes in deter i te whe

c O an d 00 . fa tor , the other INDETE MINA TE O MS R F R . 75

n a o f o ne o f the ac x n By taki g the rec iproc l f tors , the e pressio 99 may b e ma de to take the form o r . 8 0 0

F r x am ﬁnd a u o f o e ple , the v l e

7r 2 a tan a n a: ( ) , whe

This takes the fo rm 0 But

7r ; 2 w 0 (7: 2 x) ta n u: when a: c ot a;

u 2 The val e is found by A rt . 53 to b e .

5 TO eva luate a uncti o n tha t ta kes the o rm 00 oo . 7 . f f

T an m x n n a ac n c a um r sfor the e pressio i to fr tio , whi h will ss e 9—0 either the form o r . 8 (I)

Fo r x am ﬁnd a u f e ple , the v l e o 1 1 = n x l . — ’ whe lo g x w l

T 0 0 —oo his takes the form . B ut

1 en s: = — — wh 1 . lo g x x l (cc 1 ) log a: O

The a u u v l e is fo nd by Ar t . 53 to b e

EXA M P L ES.

Find the values o f the following

1 when a' tan a:

2 . s w : ec 3 c o s 7 x, a)

3 . s ee as au x t , 76 R N DIFFE E TIAL CALCUL US.

1 x — 4 . a 1 } (E T-22 ( M, m .

' 5 . a : 0 c o s ec a:

2 6 . c o s e c cc

7 . : a 1 . x 1

8 . 1 tan a s e c 2 x ( ) ,

A ns . — ’ oo . lo g ( l w)

z a — af tan as = ( ) a .

log tan 2 g: x : lo g ta n a:

2 1 — ’ s in% 1 c os a: — 13 . 2 w an cc t 1r s e c cc ,

: x 1 . A ns . 2 .

TO eva lua te a uncti o n tha t takes either o the or ms 5 8 . f f f , 0 0 0° 0 oo 1

T ak a m n unc n ch a um e the log rith of the give f tio , whi will ss e - ‘ m m 0 n n a ua b A r . F 0 0 a d c a b e t 56 . the for , ev l ted y ro this the value o f the given func tion c an b e found .

L S 7 8 DIFFERENTIAL CA L C U U .

a : 1 Ans . 1 . 11 . tan

CE tan 2 12 .

( c o t a: O

w : 0

A ns . 1 .

l 2 x : Ans . 6 . ( e a) O C A PTER IX H .

PA RTIA L DIFF ERENTIA TIO N .

In Functi o ns O s everal In e en ent ari a bles . 59 . f d p d V the pre c eding chapters differentiation h as been applied only to func n o f a n n n n a a n n tio s si gle i depe de t v ri ble . We shall o w c o sider n f r m n n n functio s o two o ore i depe de t variables .

n R n n P arti al D ere ti al Co e ici ents . b u a 6 0 . iﬁ ﬁ eprese ti g y un c n o f two n n n a ab a; and f tio the i depe de t v ri les y,

= w a u f ( ) y) ( )

di ﬁ erenti ate a u n a; a and If we ( ) , s pposi g to v ry y to i c n an b a n i o st t, we o t i dd:

diﬁ erentiate a u n a and a: ma n If we ( ) , s pposi g y to v ry to re i

c n an b a n FE? o st t, we o t i dr du du The di ff n a c o efﬁcients u are ere ti l , , th s derived, called doc dy du da p arti al dizferenti a l co eﬁici ents and are den oted 1057

3 2 3 Fo r exam le if u = w 3 v — p , + y y ,

da ” 3 w v a n a c n n 6 s a . y, reg rdi g y o st t 6a:

1 2 2 3 x 3 ardi n a: as c n an . 3; y , reg g o st t

In n a a numb o f n n n a ge er l, wh tever the er i depe de t vari bles the partial differential c oefﬁcients are obtained by supposing nl n im o y o e to vary at a t e . 80 N CA L C L DIFFERE TIAL U US.

EX A M P L ES .

du — —- w l y éE

— — du Ca Ca u : z z a: x — (y y) , + dx By dz

— 3 . 3 x z y ),

6 ” du du

6x dy

“ - e log e N w s,

2 ’ e e ” 2 ez +y sin a: ( + y) .

7 . u an z t ), sin 2 wgg+ sin 2 y§§ sin 2 z

' ‘ P ti a l D e enti a l o e i ci ents O H i h 6 1 a r r C er r ers . B . iﬁ ﬁ f g O d y u c c n a n a n n n n s essive differe ti tio , reg rdi g the i depe de t variables as a n nl o ne at a m ma b a n v ryi g o y ti e, we y o t i

E n n n m 2 , 2 ’ 3 ’ " 6513 8y 6x Cy

i f n a u c x n u If we d f ere ti te with respe t to , the this res lt with 62u c Ob a n 5? c n respe t to y, we t i whi h is writte dydw 3 6 1 S m a u o f ucc n a i il rly, 2 is the res lt three s essive differe ti 8y823

n c cc an d o n e c to . tio s, two with respe t to , with respe t y It will n o w be shown that this result is independent o f the order of

diﬁ eren i i n these t at o s . P A R TIAL DIFFEREN TIA TION. 81

z 2 3 3 3 d u d u d u d a d a T‘ a 18 h t ’ ’ 2 e dydfv da dy agar: 833 8n se dy

2 Given u v 6 . f ( , y) to p ro ve tha t 8a;

n a: a n c an in a Supposi g lo e to h ge ( ) ,

Au a: Av =fl , y) y) Aw Am

No w supposing y alone to change in (b)

w Aw + A _ w A w _ f( + 9 y y) f( a y + y) y) +f( ) Ay Aw

b ﬁnd Reversing the a ove order, we

Au w A f ﬁ , y r) y) A?! Ay

w Aw A — -r f( + , y + y) y) f( , y) Aa‘ Ay

Hen c e

T be n u ma Ace an d A ma be a his i g tr e, however s ll y y , we h ve for the limits of the above

z z d u d u

ayaw decay

T nc that the o r er o di erenti a ti o n i s imma 6 3 . his pri iple, d f ﬁ i a l ma e x n an num r o f f n ter b b a n . , y e te ded to y e dif ere ti tio s 3 3 d a d u Thus , 2 dyda: (9908w

3 d d a _ _ ’ div daf dy It is evident that the principle applies also to functions of

o r m three ore variables . 82 A L DIFFERENTIAL C C UL US.

EX A M P L E3 .

2 2 8 “ d a V i in ac o f u n er fy ‘ e h the fo r followi g equations 6n Goody l m ? 1 . u o 1 . u i y g ( y) 3 . s n (my ) — _ a v by 2 . 4 . . “ ay baz

2 2 2 z $ 11 d a d u da u 5 . S OW tha1: w + If ’ h f y w + y da docdy 8a:

z 2 2 f d u d u du x ' ‘ 6 . i s + u y 2 away dy 6y 3“ 8 2 2 7 (1 3 xyz x y z ) u dv dydz

6 ' 6 “ 2 2 g 2 2 g 2 2 g ; ; 5 8 . u z e z x e x e e e 3 y y , f’ 2 (iaf ély dz

u : sin 2 n z 9 . (y + ) si ( 3 d u 2 c o s (2 v + 2 y + 2 z) acr3as

To ta l D erenti a l. u a unc n o f m 6 4 . iﬁ If is f tio two or ore a an d all a at am m c an in u vari bles , v ry the s e ti e, the h ge is to ta l increment and n n ma to tal di l c alled the , if i ﬁ itely s ll, the f f erenti a l o f u . This total differential o f u may be o btained by the u sual f f n a n u n n a a in A 2 formul ae o dif ere ti tio , si g differe ti ls s rt. 8 .

r xam u Fo e ple, s ppose

3 2 2 x w i t y 3 y .

n a n b a: an d a ab Differentiati g, reg rdi g oth y v ri le,

3 ? du d (a; y) d (3 defy ) “ i ” ’ ’ M y ydw) 3 w de ) 3 r d tv ) 3 z 2 f x dy 3 v ydtc 6 x ydy 6 xy dv

f 2 3 2 v v (3 mg 6 xy ) dx ( 6 y)dy. A D TOT L IFFERENTIAL . 83

2 2 3 2 3 v —6 cv i and v — 6 x y y gg, y

du — da— Hen ce du dm+ Cly da; dy

This expression for the total differential holds for a ny func

n o f a ab u v . tio two v ri les , f ( , y) Fo r if diﬁ erentiate u a n u n ff n a as , we this eq tio , si g di ere ti ls

n xam ma a ran m in i n the pre c edi g e ple, we y r ge the ter s two

u co n a ni n dx an d d c a u gro ps t i g y respe tively, so th t the res lt will be of the form da P da; Qdy (2)

No w a: a n a b n c n an 2 b c m if lo e v ries , y ei g o st t, ( ) e o es

d a Pdfv n Q2? P w , givi g . 890

a n a a; b n c n an 2 b c m If y lo e v ries , ei g o st t, ( ) e o es

6 d u d n Q y, givi g . y 22 Q

Sub u n in x n P an d a stit ti g (2) these e pressio s for Q, we h ve i s du da: dy 6a: dy

S m a u w z ma be n a i il rly, if f ( , y, ), it y show th t

— — — dx + dy + dz (4) d a

The u c n a c ma be ac a 6 5 . res lt of the pre edi g rti le y re hed lso

n The to tal d erenti a l o a uncti o n O i n the followi g w ay . ilf f f f s ever al i ndep endent va ri a bles ASS the sum Of i ts p arti al d izferen n o he v i l tials ar ising fr o m the sep arate vari a tio f t ar a b es . f L Au du n o a nc m n an d n a o f u et . , , de te the tot l i re e t, differe ti l A u A u d u d u a a nc m n and n a z , y , x , y , the p rti l i re e ts differe ti ls, a: n when a d y vary separately .

Let = x u f ( > y), , u x “ " Am f ( 1 )

u 84 DIFFER ENTIA L CA L CUL US.

A u u ,

' A u u y

A u u u .

' nc Au A u A u He e x y .

No w Aw A and c o n u n A u b c m if , y, seq e tly z , e o e nitel ma a y s ll, we h ve = mw dﬂ 4ﬂ % % ' nc m f u si e the li it o is u .

( u m a Cl u — da d u We y write , , , 2)a:

9 n 2 oc givi g d dy. dv dy

Th e proc ess above m ay be exten ded to fun ction s m a a ore v ri bles .

EX A M P L ES.

F n a in Ar 4 a f n a o f u in ac o f i d s t . 6 the tot l dif ere ti l e h the

o n 4 . an d o a a A rt. 6 f llowi g, sh w th t it grees with

2 z : 1 . u a s: 2 b.r c du 2 a cv y + y , ( +

= 2 . i t du u duo

4 w2 a y 4 § — u : l 2 doc a rt . 3 . o + tan y

F n n f u in ac o f o n an d i d the total differe tial o e h the foll wi g,

4 . sho w that it agrees with A rt . 6

4 .

du 2 (ha + by +fz)dy+ 2 (ga:+fy + cz) dz.

y” yz “ 1 m z cv za; lo a d m l o a d z . 5 . u da x d , (y g y y g )

2 2 2 6 . u tan a: n t n z ta y a , sin 2 a' si n 2 y sin 2 z/

86 DI ERENTIAL CALC L FF U US.

E A M L X P ES.

' By means o f (1) determin e which o f the following expres sion s are exac t di fferentials

2 2 1 . Sp 2 dv 90 2 v Cl ( y y ) ( y) y.

4 er w — — . r< y wy 1> dyi

Show that c ondition (1) is s atisﬁed by the an swers to Ex am 1 Art n 3 . 65 a d c ndi n 2 b an ples , , ; o tio s ( ) y the swers to Ex m 4 A r a 5 t 65 . ples , , .

i h fl n D erenti ati o n O an Im li cit Functi o n . T e di ere ti l 6 7 . fi f p a c oeffic ien t of an implicit fun ction may be expressed in terms o f a f n c f n p rtial di f ere tial oe fi cie ts .

Su n nn u n < w Le a d a: c c b a 0 . t ppose y o e ted y the eq tio 5 ( , y)

u n m mb ua n . T a represe t the ﬁrst e er of this eq tio h t is,

( 17 O 1 u P0 ) y) ( )

F m Ar 4 a iff n a o f u t . 6 a ro (3) , we h ve for the tot l d ere ti l ,

du du dot dy 6a: dy

B ut b u a a zero a a c n an and e y is lw ys , th t is , o st t ; ther u m e z nc fore its total di fferential d u st b ero . He e

6a: dy

Ga:

631

Fo r xam u as in Art . 31 e ple, s ppose, , f 2 z 2 2 2 0 a y b v b . T L DIF T P A R IA FEREN IA TION. 87

f 2 "’ 2 2 2 a b x a b y ,

z 2 2 b x b x ence b 2 H y 7 2 2 2 a y “3,

913 Derive by (2) the expression s fo r in the examples in d” Ar 1 t. 3 .

’ Extensi o n O Ta lo r s Th eo rem to uncti o ns O w inde 6 8 . f y f f t o ’ n ent f m p e d va ri a bles . I we apply Taylor s Theore to

w h k f ( . y ) .

a n x as nl a ab a reg rdi g the o y v ri le, we h ve

w h Ic = w k h 7s f . f + r . ( r ) ( , y ) i e y )

2 2 71 6 h 1

N x an n x a di n as nl ar ab ow e p di g f( , y reg r g y the o y v i le,

2 2 6 76 6 x 70 = w le— ” f( , y ) f( ) y) i ft ) y) y) ag

Substitu tin g this in 8 h 7c = w h to re , y ) f( , y) rd , y) mr y) 55 gi7 32 82 91 2 w f( r r) 79?

This may be expressed in the symbolic form thu s 8 + r) 55

d o n 88 DI FE ENTIAL L F R CA C UL US .

k be x an b B n m a T m where + is to e p ded y the i o i l heore ,

a a as h — an d k — m o f b n m a and if were the two ter s the i o i l, the dx dy m x u n a a a . res lti g ter s pplied sep r tely to f( , y)

’ Ta lo r s T heo rem a li ed to uncti o ns number O 6 9 . y pp f f B m a o f i ndep endent va ri a bles . y a method si ilar to th t the pre c eding article we shall ﬁn d a ‘ ‘ k z k l x z y l r ) + + + f( r y) ) w dy

hi k 9~ z v z + + re , y. ) dx dy

n x n an num o f a a Thi s expa s ion may be e te ded to y ber v ri bles .

90 DIFFERENTIA L CA L C UL US .

7 1 . i s m m n c a in dff n a It so eti es e ess ry the i ere ti l coefﬁcients ,

2 3 dy d y d y ’ ’ 2 ’ dx dx d

n uc a new a ab in ac o f x o r 2 b n a n to i trod e v ri le z pl e y , ei g give

function o f the variable remo ved .

T are c a ac c n as a c o r x . here two ses , ordi g z repl es y

f 3 dy d g d y Fi rst. To ex ress i n t ms 0 7 2 . er p ’ z’ 3 ’ f dx dx dx

where i s a iv en uncti o n O z . ? y g f f Ex

3 F r x m u z . o e a ple , s ppose y

dy dz Then 3 dx dx

d2 y 6 3 2 z 2 dx doe

2 f dz d z 2 d z + 3 z _ 2 g dx dx din

m a ma b e x in m Si il rly, y e pressed ter s

and x .

It is to b e n otic ed that in this c ase there is no change o f the

nt a ab c ma n x . i ndep ende v ri le , whi h re i s

2 3 3 i 1 — i terms o . To ex ress n 7 3 . Seco n d p , 2 , 3 , f 513 $53 tl3

where x i s a iven f unctio n Of z . 2, gig, g 3z

This is c alled cha nging the i ndep endent va ri a ble fro m x to 23.

” Fo r x am u x z . e ple , s ppose

1 A 22 B rt . y ( ) , dx dz dx CHANGE OF VA RIAB LE.

z dx 3 dz

i’ Cl d d d d dz 2 y y y B 1 Art . 2 y ( ) ’ d? dx dx dz dx dx

F m a ro ( ) ,

2 d y — 1 f (Tof 9

3 2 d y d d y dz S m a i il rly 3 2 dx dz dx dx

-5 —6 F m b _ 6 z 10 z ro ( ) 2 2 + Clz dx 9 dz

3 (P 1 d g _6 y _8 l 0 z s g + dd 2 7 de

EXA M P L ES.

Change the independent variable from x to y in the two lowing equations :

2 3 df dy d y . g ° 3 2 3 2 dx dx do: dy 0131 92 DI E ENT AL CAL C UL FF R I US.

Change the variable from y to z in the two following equa tio ns

3 d z n z 1 —— _ A s . ( + ) gx

Change the independent vari able from x to z in the foll owing equations

2 2 : : : 5 0 x 4 z . A ns . z + 0 . 2 , y dx x dx

f d 2 x d _ g y y _ 6 . 0 x tan z .

E 94 DIFF RENTIAL CALCUL Us .

6 The 7 .

This cu rve may b e c onstr ucted from the c ircle ORA

a u a b a n an ab c a MR and x n n (r di s , ) y dr wi g y s iss , e te di g it r to P by the c o ntru ctio n shown in the ﬁgu e . m n n Th e equ ation above m ay be derived fro this c o structio .

Th e axis o f X is an asymptote .

? 3 The a x 2 00 . 7 7 . EP RESE TA TI F O R E R N ON O U V S. 95

The Ca tenar 7 8 . y,

5 y g(6

This is the c urve o f a c ord o r chain su spe nded freely between n two poi ts .

’ The P dm bo la re erre 7 9 . f d to Ta ngents at the Extremitzes of

‘ the L atus R ectum xi i } , y a .

L ' = 0 0 L a .

' The n LL a u c um m n F li e is the l t s re t ; its iddle poi t , the fo cu s ; OFM is the ax is o f the parabola ; the middl e point o f OF A x , , is the verte . S 9 6 DIFFERENTIAL CALCUL U .

= The Hyp o cycloid fi + fi fi

This is the curve d e s c r i b e d by a poi n t P i n the c ircu mferenc e o f

c r c PR as the i le , it rolls withi n the c ir c u mfere n c e o f the ﬁx ed c ircle ' B A . ra A , whose di u s a u , , is fo r times that of the m for er .

The 1 . ur e 8 C v ,

The equation is that o f the ellipse

x n n n m 2 L with the second e po e t cha ged fro to 3.

98 DI ERENTIAL US FF CALCUL .

i 8 5 . n n 1 A rt . 83 a If 7 , ( ) , we h ve

l - % i z a The Semi Cubi cal P ara bo la a m o r a , y , g m.

— POLAR CO ORDINATES.

Th ir le r sin e c a . 86 . C , 0

The c c OPA am a an n . n a n ir le is (di eter, ) t ge t to the i iti l li e

X at . O the pole, O

3 The S ira l o Archime es r a 0. 87 . p f d ,

A um n In this curve r is proportional to 0. ss i g w n 6 2 7r n he , the 0 P = 0 A OP = 1 0 A OP ==5 OA 1 4, , 2 , 5 , REP RESENTA TION OF OUR VEs . 99

The dotted part o f the curve c orrespon ds to megati ve values

The Lo arithmic ira l 88 . g Sp , r

S a n o m A t rti g fr , where

and r = i r n c a , i re ses with 0 ; but if we sup 0 n a r pose eg tive, de c reases as 0 numerically n n = i creases . Si c e r 0

n n 0 —cc o ly whe , it follo ws that an inﬁnite nu mb er . of retrograde revolutio ns from A is required to reach the pole O. A property o f this

S a a a c m m a pir l is th t the r dii ve tores , ake c n an an cu o st t gle with the rve . 100 CA L C UL s DIFFERENTIAL U .

2 The P r l r . 89 . a a bo a a 3 9 0 , 3

The initial line OX is the axis o f the parabola; the pole O ' cu LL a u m c u . is the fo s ; , the l t s re t

Th 2 2 e L emniscat r a o s 2 . 90 . e, c 0

T a cu o f ﬁ u his is rve two loops like the g re eight . It may b e deﬁned in c onn ection with the equ ilateral hyper b a as cu o f P o f a n cu a m 0 o n ol , the lo s , the foot perpe di l r fro P an an n b a Q, y t ge t to the hyper ol . Th e are m b a m o f loops li ited y the sy ptotes the hyperbola, making ’ TOX T OX OA a .

The lemniscate h as the following property tw o n F and F b e ak n o n ax uc a If poi ts, t e the is, s h th t

' OF : OF V2

' ' ' t n uc o f anc P F P F an n o f he the prod t the dist es , , of y poi t cu m x n c n an and ua the rve fro these ﬁ ed poi ts, is o st t, eq l to the u f sq are o OF .

102 CAL C UL S DIFFERENTIAL U .

The Ca r i o i r a i o s 9 2 . c d d, (

This is the curve des cribed by a po in t P in the c ircumfer‘ n c a c c PA am a as u n an u a e e of ir le (di eter, ) it rolls po eq l x ﬁ ed circle OA .

m be c n u c e b a n o u 0 an i n OR Or it ay o str t d y dr wi g thr gh , y l e ' i n c c OA and uc n OR an d mak n the ir le , prod i g to Q Q , i g = ' R Q R Q 0 A . m n uc n The given equ ation follows directly fro this c o str tio .

= The r a n 2 9. 93 . si C APTER XII H .

DIR EC TION OP C U R V E NG ENT A ND . TA N R M O A L.

M P T TES A S Y O .

irecti o n o u 94 . D f C rve . When the equation o f the curve is

n in r c an u a c o - na o n at an n give e t g l r ordi tes , its direc ti y poi t is determined by the angle m ade by its tangent at that po int with the ax is o f X . We shall denote this a ngle by Let P b e a po int in a c urve whose equatio n is y its - c o ordinates b eing a} : OM and

a an n y P M . Dr w the t g e t

P T and PR a a OX . , p r llel to

Then TFR (I) . No w give to a; the increment A w : MN ; then y will rec eive nc m n A = R and a an o n in the i re e t y Q , we h ve other p i t Q the

u a P . c rve . Dr w Q

T tan PR — a hen Q $32 iZ ( )

N A x be u to m n and a ac z A o w if s pposed di i ish ppro h ero , y

a ac z n l mo a o n c u will ppro h ero , the poi t Q wi l ve l g the rve i i T s o a P and P a o ac n d c o n P a m . t w rds , Q w ll ppr h ire ti its li it

Tak n the m o f m mb o f ua n a i g li its the two e ers eq tio ( ) , we have m f P TP 8 li it o Q R R 9 ,

é —g g_y m o f b n o n . li it , y deﬁ iti A ce Ola;

da: 104 CAL C UL Us DIFFERENTIAL .

Fo r xam ﬁn d c n at an n o f a ab e ple, the dire tio y poi t the p r ola 2 = y 4 a m.

div

henc e t an <1)

At x 0 a: 0 the verte , where ,

tan <1) <1;

At L w a , where ,

tan < ) 1 1 , ct

Fo r that part o f the curve

b n L as 93 n c a tan c eyo d , i re ses, b and T u ar 4) decrease . h s the p abola is more n early parallel to m OX u x n O. , the f rther it e te ds fro

L an n u btan ent Subno rma l. et P T b e S g the t ge t, n d P n ma a a N the or l, to

u at n P c rve the poi t , whose ordinate is y PM Then M T c a s ubta n ent is lled the g ,

and MN s ubno rma l c o r. the , respo nding to the point P To ﬁnd expression s for these quantities

Subtangent =M T PM co t P TM : y co t

doc

—dgl Subnormal = MN = PM tan MP N: y tan q5 y . eta)

mal. n The length PN is sometimes c alled the no r It is evide t that PN = PM sec

106 CAL C UL DIFFERENTIAL US.

i - 96 . recti o n o Curve. P o lar Oo o r ina tes D f d . By means o f the equations

a = r c 9 = r s i os , y n 0

ma x tan t in m o f r an we y e press q ter s d 0 . Thus

? t i ! ‘ r co s 9 “P s in 9 — g ge tau gt _ g ga: ga — r s in 0 + c os 0 d g aO

The angle OP T b etween the tangent and the radius vector may also b e e x

pressed . Denote

this angle by ll] .

the Let r , 0, b e c o -o rdinates o f P ; A r A r , 0 0 , the c o -ordinates b o f Q . Des cri e the arc PR about

O as a c entre . Then

= = = A A . R Q r , P OR A 0, P R r 0

u s a ac P u P R a ac If we s ppo e Q to ppro h , the ﬁg re Q will ppro h mo an n a an R b n an . re d more e rly a right tri gle , ei g the right gle We have at the limit

tan P QR

We also have

P TX : OP T P OX , E 10 DIRECTION OF C UR V . 7

P o lar Su bta n ent and 9 7 . g l Su bno rma .

u O NT b e a n If thro gh , dr w

r n cu a P OT pe pe di l r to O , is c a o la r s u bta n en t an d lled the p g ,

o la r s u bno rma l c o r ON the p , n n n respo di g to the poi t P .

T = P tan OP T a O O th t is,

Polar subtangent r tan ll! 5; d0

: R c o t R a ON O No th t is,

—l Polar subnormal r c o t tp Elz

EX A M P L ES .

ﬁn In c c r a n 0 d l an d ( ) . the ir le si , al I

and 2 9. A ns . gb

a " In a m c a r e a ( ! c n an . the log rith i spir l , show th t I is o st t

In a o f A c m r = a 0 a am z 0 the spir l r hi edes, , show th t t / ;

n c ﬁnd a u o f ' n 2 an d 4 the e the v l es ll whe 0 7: 7r. ° ° 0 5 an d 85 A ns . 8 7

n n Also show that the polar subn ormal is co sta t .

The equ ation o f the lemnis c ate referred to a tan gen t at its 2 = 2 c n r a n 2 0. F n i an d o a sub e tre is si i d t, the p l r n n ta ge t . T 3 i n . ! A s (I 2 0 ; qt 3 0 ; su btan gent a tan 2 9 V s n 2 0.

= 3 Given the equ ation o f a cu rve r a sin g; show that (I)

z In a ab a r a sec a c . the p r ol g, show th t ]; 1r 108 CA L C UL Us DIFFERENTIAL .

7 . In c a r d l co s ﬁn d t 1 and the rdioid ( q , 51, the polar n n subta ge t . s o 9 0 , n 2 A ns . <1) , ; s ubta gent 2 a tan sin 5 5 g 2‘

F n a a o f c cu m c b u a c di n 8 . i d the re the ir s ri ed sq re of the pre e g ° m n n n n 4 b a c 5 to ax . c ardioid, for ed y t ge ts i li ed the is

A ns .

i ua n a m u a n 6 and c A rt . 96 . 9 . Der ve eq tio ( ) fro eq tio s ( ) ( ) , of

- D erentia l Co e ci e t o the rc . R ectan ular Co o roli 98 . iﬁ ﬁ n f A g

In u f A r 4 n n f h na tes . o t . 9 3 o t e the ﬁg re , let de ote the le gth m m n x n f arc of the c urve easured fro a y ﬁ ed poi t o it .

n r A P A r P T 3 a c s a c . he , Q 1 Q We hav e s e c QPR

No w u A91: a ac z and n s ppose to ppro h ero, the poi t Q to approach P . Then limit se c QPR sec TPR se c

. P Q . . arc P Q . A s (18 1m1 1m1 i mi R P R Arc dw

Henc e s ec qt dcv l Ei 2 ‘ therefore tan ¢ = 1 dx V It is evident also that ﬁg da: s rn < ; c o s < ; 2 1 ’ 1 ( ) as as

It may be n otic ed that these re latio n s (1 ) an d (2) are c orrectly rep n b a an rese ted y right tri gle, whose nu as dx and d hypothe se is , sides y, n n a d a gle at the b ase (I) .

1 10 CA L C UL Us DIFFERENTIAL .

ﬁgbeing derived from the equatio n o f the give n curve y ' ' a nd a i to o n o f c n ac a ppl ed the p i t o t t ( , y ) l 93 n b , a ub u n m If we de ote this y l we h ve , s stit ti g daz eq uatio n ( a ) d ' — ' y y y ' dac fo r the equ ation o f the required tangent .

' ' n n ma a n u x n cu a Si c e the or l is li e thro gh ( , y ) perpe di l r to the

an n w a ua n t ge t , e h ve for its eq tio 1 CI) 55, a?

ﬁnd ua n o f an n and n ma to Fo r ex ample , the eq tio s the t ge t or l 2 2 2 ' m : a at n w the ci rcle 31 the poi t ( ,

2 2 2 b d f n a n x a ﬁnd Here , y if ere ti ti g y , we

I i l l / a: fl — g m c El fro whi h 7 da: y doc y

Sub stituting in we have I — ' y y

as the equation o f the required tangent .

It may b e simpliﬁed as follows

l 2 ! l _ mm a; gy ‘y ! l = l 2 l 2 2 mm yy x + y “. m 2 The equation o f the normal to the circle is found fro ( ) to be I y which reduces to

3/ A D TA NGEN T N NORMA L . 111

EX A M P L ES .

Find the equatio ns o f the tangent and normal to each o f the ' ' n u at n w three followi g c rves the poi t ( , y )

2 = h a ab a 4 am. 1 . T e p r ol y

2 a A ns . (y

2 —= h 1 . 2 . T e ellipse “2 2

l l i q z '

ns b x . A . + (y f; %2

2 The u a a b a a . 3 . eq il ter l hyper ol w

’ ' 2 ' ' ' ' A ns . a ac a cc cc w y y , y (y y ) ( )

a i n n cu a a o f an 4 . Show th t the prec edi g rve the re the tri gle formed by a tangent and the c o -ordinate ax es is c onstant 2 and equal to a .

3 2 90 In c ﬁnd ua n o f an 5 . the issoid y the eq tio s the t 2 a — x

n no ma a b a gent a d r l t the points whose a sciss is a . = — = A ns . A t a a 2 x a 2 w 3 a . ( , ) y , y + — m= = — A t a a 2 a 2 w 3 a . ( , ) , y + , y

g 8 a

6 . In c ﬁnd ua o n o f an n the wit h 3] 2 2 ’ the eq ti s the t ge t 4 a + x

n n ma a 2 a d or l t the point whose ab s ciss a is a . m = 4 = — A ns . 2 a 2 x + y , y 3 a . i

7 . In cu 1 ﬁnd ua n f the rve , the eq tio o the

r I ' y n n a n w A ns . a t 1 . t ge t the poi t ( , 2 “ ﬁ ' i 3 b y

2 2 8 . In x 2 2 w a; 0 ﬁnd ua n f the ellipse y y , the eq tio s o an n and n ma at n 1 the t ge t or l the poi ts whose abs ciss a is . = — = Ans . A t 2 w 1 2 w 2 . y , y + A t 2 = w = y + 1 , y + 2 x 112 DIFFERENTIA L CA L C UL Us .

l - l 7 In a ab o a ai a ﬁnd u a o n o f ~ the p r l 1 y , the eq ti the tan ' n at o n x " ge t the p i t ( , y ? 9 ° A ns . xx a

Sho w that i n the pre c edin g c u rve th e sum o f the interc epts o f the tan gent o n the c o -ordinate ax es is c o nstant and

equ al to a .

g‘ l’ ii In c c o m (i ﬁnd ua n o f the hypo y l id y , the eq tio the ' - — an n at n w r r r r t ge t the poi t ( , i l d A ns . xx a yy .

Show that i n the prec eding cu rve the part o f the tangent interc epted between the co -ordinate axes is c on stant an d

equ al to a .

- A s m to tes . R ectan ular Co o r ina tes . W n 1 00 . y p g d he the an n a cu a ac a m n n as t ge t to rve ppro hes li iti g positio , the dis tan c e o f the point o f c ontact fro m the o rigin is in definitely m n n c a an a m In n a . i c re sed, this li iti g positio is lled sy ptote an a m a an n c a n a other words, sy ptote is t ge t whi h p sses withi n an c n a u n o f c n ac at fi ite dist e of the origi , ltho gh its poi t o t t is n n n an i fi ite dista c e .

m u n n n 1 F a a 1 Art . 99 w e 1 0 . ro the eq tio of the t ge t ( ) , ﬁnd n c o n c c - na ax for its i ter epts the ordi te es,

' " n c o n a 2 I ter ept X ' dy

' , ' d “ - y- Interc e o n Y : x o pt y ’ dx

' ' o f n c n x 00 o r = oo If either these i ter epts is ﬁ ite for , y , n n n n b e an a m the c orrespo di g ta ge t will sy ptote . The equ ation of this asymptote may b e obtained from its n o r m o ne n c and m n a u two i tercepts, fro i ter ept the li iti g v l e

114 DIFFER EN TIA L CA L C UL Us .

T are n a m o here the two sy pt tes , who se equ ations are é y = i ca a

The a n no n n b anc ellipse, h vi g i ﬁ ite r hes, can have a m sy ptote .

1 A s - 03 . m to tes P a ra llel to the Co o r inate A xes y p . n i d Whe , n u a n f o c u a; 00 a n a u as the eq tio the rve, gives ﬁ ite v l e of y, a n , a ua o n o f an a m a a y the y is the eq ti sy ptote p r llel to X. A nd n = oo m= a n at = a n whe y gives , the is a asymptote a a p r llel to Y.

1 s m t 04 . A o tes b Ex a n i y p y p s o n . Frequ ently an asymptote ma b e y determin ed by solving the equation o f the c urve for a:

o r and x an n c n m m r y e p di g the se o d e be . Fo r xam ﬁnd a m e ple, to the sy ptotes of the hyperbola

— a ) 2

A s 9: nc a n n cu a i re ses i deﬁ itely, the rve pproaches the = 993 m d: a . y , the sy ptotes a

- A s m to tes . P o lar o o r in s F m C a te . f 1 0 5 . y p d ro the ﬁgure o n m A rt . 97 n a a a a ub an , it is evide t th t for sy ptote, the pol r s t

n T h as a n m as OP n n nc a ge t O ﬁ ite li it, is i deﬁ itely i re sed .

i 0 T a n r — h as a n m fo r r : 00 an h t is, whe g ﬁ ite li it , there is m n m and a a a at a a c r . sy ptote th t dist e fro the pole, p r llel to “ 2 9 an 7 — v an d If the dist ce is positi e, it is to the right, if 37' n a o f k n in d c n o f eg tive, to the left, the pole, loo i g the ire tio n n the i ﬁ ite r . MP T TE 11 A S Y O S. 5

Fo r xam ﬁnd 1 06 . e ple, the asympto tes o f the curve

r a tan d.

2 a s e c d , dd

and the subtan gent

2 a sin d.

77 W h n d j : e ’ 2

a r 00 we h ve ,

n n n a d the su bta ge t a .

There ar e tw o asympto tes perpen dicu lar distance f a m o n ac o . fro the pole, e h side it

EX A M P L ES .

Investigate th e followin g cu rves with referen c e to asymp totes :

3 x

7 1 . A m o so . y 2 2 sy pt te, y 90 3 “

3 2 3 = x - A m o az = 2 . 6 x . 2 Y y sy pt te, + y .

L h 2 A m = 3 . T e c o ac 2 a 7 iss id y sy ptote, .

3 3 3 A m a; 4 . £13 a 0 y . sy ptote, g .

2 3 3 — 2 a = a — A m o oc = 2 a = 5 . r a . x a ( ) y sy pt tes , , + i y.

—3 3 6 . x 3 61907 m a: a ]?e “ , 0 . y . 1&c Asy ptote y ( Sub stitu te y we i n the given equatio n an d in the “l “ V V x r n f r e p essio s o r the inte c epts . ) i The c r o r . 7 . re ip c al Spi al r d

A m a a OX at anc a ab sy ptote p r llel to , the dist e ove . 16 DIFFERENTIA L CA L 1 C UL US.

= r a s e c 2 d. There are fou r asymptotes at the s ame distanc e 3from the ° an d n n 45 c OX . pole, i li ed with

a Th r o l a no a m e pa ab r There is sy ptote . 1 — c o s d

( r a ) sin d b.

n m a a OX at i s an c b There is a asy ptote p r llel to , the d t e

above .

n r a (se c 2 d + ta 2 d) .

1 r o m a a d at anc There a e tw asy ptotes p r llel to , the dist e 31 a o n eac h side of the pole .

118 DIFF E ENTIA L CAL C L R U US.

F m c n u a n ro the se o d ﬁg re , we see th t whe the curve is c o n

a d o wnwa r tan ( ) ecrea ses as a: n a and c ve d , I d i cre ses , therefore

z d ?! a 0 . th t is ,

1 1 A P o int o In ea' io n f a cu 0 . o a n P f ﬁ rve is poi t , where the c u a u c an c u o n o ne o f n b n rv t re h ges , the rve side this poi t ei g

c o nc a . u a and o n ve pw rd ,

c o nc a o n the other, ve d w

a . n b 1 c A rt. 0 9 w rd He e, y , z d y n f inﬁ x i at a o e o n , poi t t dic an n a ch ges sig ; th t is ,

’ 0 0 1 00 . 2

It is evident that the tan gent at a point o f inﬂex io n interse cts the c urve at that point . Find the point o f i nﬂex io n of the c urve y (w and the direction o f c urvature o n each side o f it .

2 d y Here 6 a: 1

u n ua z a fo r u n o f P tti g this eq l to ero , we h ve the req ired poi t 2 d d y : 1nﬂ x 10 n a 1 . w 1 0 a nd 1f x 1 0 . e , If < , < , > , > E£2 w

n a o n ft and co n Henc e the c urve is c onc ave dow w rd the le ,

u a o n ri o f o n o f inﬂex io n . c ave pw rd the ght , the p i t

EX A M P L ES .

inﬁe x io n and d c n o f c u a u Find the points o f , the ire tio rv t re , o f the three following c urves

f — 2 - 3 h ur e a a T 2 a . 1 . T e c v g 4 3

c nc a d n a o n o f s A ns . o ve ow w rd the left thi P OIN TS OF I NFLEX ION. 119

The witch y

0 n n a n a A s . :t 5 c o c ve dow w rd these $3 n o nc a u a u o f poi ts , c ve pw rd o tside

3 x The c u rve y 2 2 m 3 a

9 a 3 a O 0 3 a — c nc a u , ( ) , o ve p 4

a o n o f s o in o n a b t n w rd the left ﬁr t p t , d w w rd e wee

ﬁ and co n u a b e n c n a nd rst se d , pw rd etwe se o d

r and n a d o n n . thi d , dow w r the right of third poi t

Find the points of inﬂex io n o f the c urve

A 718 . a: 21:

4 ° 4 F n n a % 5 . i d the poi ts o f inﬂex io n o f the c urve a y

cc : A ns . rl: gV2 7 C A PTE H R XIV .

C U R V A T R E R DIU S F C U R V R E EV L U . A O A TU O U TE .

A ND INV O LU TE.

m i n a ai e ni tio n o Cur va ture . a n 1 1 1 . D ﬁ f If poi t oves str ght

n di c o n o f mo o n am at o n o f li e , the re ti its ti is the s e every p i t its

c u bu t a h a c ur n a c o n nua o rse ; if its p t is ved li e , there is ti l

o a s m n T an c ha nge o f directi n it oves alo g the c urve . his ch ge urv u o f dire c tio n is c alled c a t re . The direc tion a t any point being the s ame as that o f the tan

n at a o i nt the c u a u ma b e d m n b c m a g e t th t p , _ rv t re y eter i ed y o p r i ng the line ar mo tion o f the point with the s imulta neo us a ng ular u un mo tio n o f the tangent . The c urvat re is either ifo rm o r

va riable .

n o rm Curvature . The cu a u un m n as 1 1 2 . U if rv t re is ifor whe ,

ua a c an n u n t o u ua the point moves over eq l r s , the t ge t t r s hr gh eq l

t n m a u b an c b b tan a ngles . It is he e s red y the gle des ri ed y the b n r ge nt while the point des cri es a u it of a c . s P Suppo se the poi nt P to mo ve in the curve A Q. Le t A

i n a n c u f m an x n A a nd denote its d s ta c e lo g the rve ro y ﬁ ed poi t ,

= TX a n ma let qt P , the gle de by the tangent P T with the T n as ﬁx ed line OX . he the b arc P po int des cri es the Q, no t b A s which is de ed y , the tange nt turns thro ugh the T n A . angle QRK or 4) he , if un m the c urvature is ifor , it A d) is equal to . A s

u n The circle is the only curve o f uniform c urvature . S pposi g

P f a c if d a a CP a nd C and A Q an arc o c ir le , we r w the r dii Q, n o f a u n an P C let r den ote the le gth the r di s , the the gle Q

= P P an P C a is As r ) . QRK A¢ ; b ut are Q C >< gle Q ; th t , c

122 DIFFERENTIAL CA L C UL Us .

A b nt c an n a: and e a lso , y i er h gi g y , w h ve

2 dy

m m mo n which is so eti es the re c o venient ex pression .

A s an x am ﬁnd the a u f u e ple , r di s o c urvat re o f 2 3 c ubical parab ola d v x .

dy d g 3 Diffei entiatin __ _ i g , z da: % dcv 2 2 0t 4 01 51 0 Substituting in we ﬁnd

EX A M P L ES.

Find the radius o f curvature o f the following curves :

2 2 a = 4 x _ . a a a a . Ans . 1 . The b y p r ol p 3 s in ll)

2 l m . A ns The equilateral hyperb ola ﬁ y a . p

0 v__ The 53+ _ 1 ellipse 2 2

What are the values o f p at the ex tremities o f

' and mi no r ax es 9 A ns

1 a 0 u t n b . 4 . The c rve , the poi t ( , )

A ns . p

Th cu lo s ec at. 5 . e rve y g

2 }‘ 5 h a ab a 51 . 6 . T e p r ol + y RADI US OF OUR VA TURE. 123

— a 2 c atenary y - (e e 2 a

ll g a . 3 a . hypo cyc lo id a A n s . p ( rg ﬁ

2 z “ 6 cur a t a m T at n 0 0 and a ve f/ , the poi ts ( , ) ( , = = ns . A p ga nd p a . t z M . c issoid y A ns . p — ' 2 3 (2 a ct )

i u s o ur tur ’ i n P o la r - r in R a C va e CO a tes . 1 1 5 . d f O d 1 0—8 1 A rt . 1 14 us x i n m o f r ( ) , p , let e press p ter s d qﬁ

From (3 ) A rt .

F m c A rt . 9 6 ro ( ) ,

) 9 4 ‘PI

F m b A rt . 9 6 ro ( ) ,

- ' 1 tan } 0 1 ‘l’ tan 4 g?

n a n Differe ti ti g ,

u n Sub stit ti g ,

Hence 124 DIFFERENTIA L CAL C UL s U .

EX A M P L ES .

Find the radius o f c urvature o f the following curves :

; 1 . The c c r a sin d. ir le A ns . p g

a" 2 2 . Th a m e c a r e s . log rith i spir l An p r a .

3 . Th a f A m e o c r a d. A ns . spir l r hi edes p 2 2 r 2 a

2 r : a 1 c ardioid ( c o s Ans . p 3

3 r a s i n 9 8 c urve Ans . p a 3 4 3

2 9 3 0 . a ab a r a s ec Ans . 2 a sec p r ol 2 p 5

2 2 “ m n . . h A s 7 T e le nisc ate r c o s 2 d. p g

- h en re o u L t at 6 . o o r ina tes o t e C t C rvature . e b e 11 C d f f , y the - na o f P an co ordi tes , y n o f the cu A B poi t rve , and C the c orresponding P c e ntre o f c urv ature . C is then the radius o f

u a u and n ma c rv t re , is or l to the c urve . Draw also the tang ent P T . = Then CP p

a ngle P CR P TX :

- Let a B b e c o na o f C . , , , the ordi tes = — = P OL OM RP , L C M + R C

a = m— sin s = co s 1 th t is , u p q , B y + p ¢ ( )

x and i n m o f 90 and a b 2 To a B Art. e press , ter s y, we h ve , y ( ) 4 9 8 , and A rt . 1 1

126 DIFFERENTIA L OAL OUL Us .

Sub u n in 2 A rt . 116 stit ti g ( ) , we have

a z : 3 at 2 a , B

E m na n m a li i ti g , we h ve fo r

ua n o f o u the eq tio the ev l te , 4 2 — 3 a a 2 B ( a ) . 227

This c urve is the semi c ub c a i l p arab o la . The ﬁg ure

o o m and o sh ws its f r p sitio n . F is the fo cus o f the given

parab ola .

OF .

1 1 P ro erties o h 9 . p f t e Invo lute a nd l Evo ute . Let u s return to u i 1 a o n A rt . 1 16 the eq t s, ( ) ,

a a: si n p d , 3 COS 5 . 3/ p 9

ff i in i n a nc 2 Ar . 98 Di ere t t g w th refere e ( ) t ,

1 A rt . 1 14 a ( ) , we h ve do d a: de s 1n (I) p c o s d) ds ds ds

d l ( _c_y_ 19 0 0 8 96 p s in (I) d s ds ds

Dividing (b) by ( a ) El—B=

d o.

' If oS denote the angle made with the ax is o f X by the tan

n to u n b 1 A rt . 94 ge t the evol te , the , y ( ) ,

d 7 B r_ 1 . tau nt . da

T a an n o u is n c u a to h t is , the t ge t to the ev l te perpe di l r the

n n an n to n u . In d a tan c orrespo di g t ge t the i vol te other wor s ,

n n F . P n to o u at a C i Art. C ge t the ev l te y poi t l ( g is l l , the EVOL UTE A ND IN VOL UTE.

A a n o m a a nd b A r . 1 19 1 20 . t g i , fr ( ) ( ) , ,

' where s denotes the length o f the arc o f the evo lute me as ured

o m a x n . n fr ﬁ ed poi t He c e ,

' ds —dp ' = j : and therefore As :t Ap ds ds

T a i s diﬁ ere nc e b n a n two a i o f c u a u h t , the etwee y r d i rv t re P 0 P 0 ua c n n n u are o f 1 1 , 3 3 , is eq l to the orrespo di g i cl ded the

u . e vol te CI C3

F m the two o f A . 1 19 and 120 1 21 . ro properties rts , it fol lows that the involute A B may b e des cribed by the e nd o f a u F m t string u nwound from the evo l te EX . ro this proper y the wo rd evo lute is derived . b e n c a a c u h as n o ne u b ut an It will oti ed th t rve o ly evol te ,

n n numb o f n o u a s ma b e n b a n the i ﬁ ite er i v l tes , y see y v ryi g u u ar length o f the string which is unwo und . S ch c rves e c alled parallel curves .

EX A M P L ES .

ind the c o -ordinates o f the c entre o f curvature o f the ' ' v cubic al parab ola gf at . 4 _ 5 M a y 9 y A ns . a , 2 B 4 6 a y 2 a

Find the c o -ordinates o f the c entre o f curvature o f the

c atenary y : g(e e

= — / 2 — 2 = A ns . d x \ OL 2 . g y , B y

- 3 . F n c o d na o f c n r o f c u a u and i d the or i tes the e t e rv t re , the 2

u i n f o u o f £ i : 1 at o . eq o the ev l te, the ellipse 2 z a b 128 DIFFERENTIAL OA L OUL Us .

l l Show that i n the p arab ola ai we have the rela = tion a + B

F n c o - na o f n f u a ur and i d the ordi tes the c e tre o c rv t e , the ii - 3 3 ua n o f u o f c c x a . eq tio the evol te , the hypo y loid k y = A ns . a B y

n ua n o f u a a h b a 6 . Give the eq tio the eq il ter l yper ol show that 3 (y x) a B 2 3 2 2 a

Thenc e derive the equation o f the evol ute — ? (a 2 a

1 30 DI E ENTIAL OA L O L FF R U Us .

W hen the o r er o co nta c i s n 1 23 . t eve the curves cro ss t d f , a the o int o co nta ct but when the o r er i s Odd the do no t p f ; d , y

cro ss .

Fo r a c n a o f o n i c m F . 1 o t t the ﬁrst rder, it is evide t fro g ,

A rt . 122 a u o f P an d P c u o n , th t o tside I 2, the dotted rve is the m a cu . nc n tw o o n s e side of the other rve He e, whe the p i ts c o nc m n c n ac cu n o t c i ide to for the poi t of o t t, the rves do ross n at that poi t .

Fo r n a o f n a c c c n o m Fi . 2 o t t the se o d order, it is evide t fr g ,

A r 22 h n t . 1 a P c nc P cu c at , t t whe 3 oi ides with , the rves ross n o f n the poi t c o tact .

Fo r o n a Fi . A r 122 a c c 3 t . a t t of the third order, g , shows th t u no t t n f n the c rves do cross a the poi t o c o tact . m n n n u Si ilarly it is evide t that the propositio is ge erally tr e .

n A n b e ma 1 24 Oscula ti Cur ves . s a a n c a . g str ight li e de to

a u o n n ta n ent h as n a a p ss thro gh ly two poi ts , the g ge er lly

n f n r u c o tact o o ly the ﬁ st order with a c rve . The c ircle havin g the closest c ontact with a cu rve at a given

n o u i n i A s a c c c an b e ma poi t is c alled the sc la t g c rcle . ir le de a o u n n cu a n c c has to p ss thr gh o ly three poi ts, the os l ti g ir le n n ge erally c o tact of the secon d order . Th e parabola of c losest c ontact is likewise c alled the o scu

n A n m o u la ti g p a ra bo la . s a parabol a c a b e ade to pass thr gh u n cu a n a ab a h as c n ac fo r poi ts, the os l ti g p r ol o t t of the third order .

Th e c oni c of closest c ontact is c alled the o scula ting co n ic . n n m u n A s a c o ic ca b e ade to pass thro gh ﬁve poi ts , the os culating c on ic h as c ontact o f the fourth order .

n m r 12 a o scul atin circl e and It is evide t fro A t . 3 th t the g

n o n u at o n o f c n ac o s culati g c ic c ross the c rve the p i t o t t, while n the tan gent and o s culating parabola do o t .

Exce tio n a l P o ints A u an n h as n 1 25 . p . ltho gh the t ge t ge er a c n ac o f r ma at xc na o in lly o t t the ﬁ st order, it y e eptio l p ts of f a c urve have a c ontact o a higher order. ORDER OF OONTA OT. 131

Fo r xam nc an n at a n inﬂexio n c e ple, si e the t ge t poi t of rosses

u r o o m A r 12 o r o f c o n ac c t . 3 a the ve, it f llows fr , th t the der t t

u n n t o n nﬁ xi n an n h as m st be eve . He c e a a p i t o f i e o the t ge t c n ac at a n o o t t of le st the sec o d rder .

Th e cu a n c c c h as n a c n ac o f os l ti g ir le , whi h ge er lly o t t the

c n rd r h as a n f m x se o d o e , higher order of c ontac t at p o i ts o a i

mu m o r m n mu m cu a u as fo r xam c o f i i rv t re, , e ple, the verti es n m f an ellipse . It is evide t fro the symmetry o the ellipse

nc c a no c rc an n at with refere e to its verti es, th t i le t ge t these

n u c cu a n f n a . n c poi ts wo ld ross the rve t the po i t o c o t ct He e, — b A rt . 123 o f c n ac o dd at a . y , the order o t t is , le st the third

A na l ti ca l Co n itio ns o r nt t 1 2 o a c . 6 . y d f C

Let = 93 and y M ) .

be the equations o f two curves hav ing two c ommon points n P a d Q.

: : 71 . Let 0 M a , MN

T n d a and a h a h . he qt ( ) d( ), ¢ ( ) ¢( )

’ Expanding each member o f this equ ation by Taylor s The o rem h3 «0 + E4) (a )

h2 h3 0 - 0 ( 04 5 3 132 DIFFERENTIA L A C L C UL Us .

S nc a a ro m 1 a n b h i e d ( ) we h ve f ( ) fter dividi g y ,

< a > +

n a o ac P h a r ac z r an d a at Whe Q ppr hes , pp o hes e o, we h ve the 11m 1t WW)

Hen c e the c o n ditio n s fo r a c o ntac t o f the ﬁrst o rder at the o n a: a are p i t , “ a M ) N ) , WM)

A 1 27 . a n u u r v g i , s ppose the two c ves ha e a c ontact o f the s at P and an c o mm n n ﬁr t order other o poi t Q .

A s b M a M h O N . efore, let ,

' S n c a and ( a a i e d ( ) p ( ) ( ) ,

2 we a m 1 A r 12 n t . 6 a b h h ve fro ( ) , fter dividi g y ,

1 h 1 u h m > - 5 a 54 l [5 9 ( ) + (a ) +

W n a ac P a at m n h 0 he Q ppro hes , we h ve the li it, whe ,

W W) .

Hen c e the c o n ditio n s fo r a c ontact o f the sec ond order at the

n a: a ar e poi t ,

( a )

134 DIFFERENTIA L OA L OUL Us

Substituting (4 ) an d (5) i n

Hen c e d = x b y +

2 dx

In x n x _ to u a these e pressio s, , y , , , refer the e gig gzg q

ti o n o f the c ircle ; bu t sin c e the o s c u latin g c ircle by deﬁnitio n h as c n ac o f c o n n cu o t t the se d order with the give rve, these qu antities will have the s ame valu es if derived from the n f n equ ation y at the poi t o c o tact .

B c m a n 7 an d 8 x o n fo r a y o p ri g ( ) ( ) with the e pressi s , B, an d in A 114 116 n a c u a n p, rts . , , it is evide t th t the os l ti g o f u u c ircle is the s ame as the circle c rvat re .

M, 1 3 0 A t a o int O ma ximum o r minimum curva ture the . p f , hi o scula ting ci rcle ha s co ntact Of the t rd o r der .

If we regar d equ atio n (8) in the pr ec eding article as re

rri n to n c u a as a c n o n fe g the give rve, we h ve o diti

fo r a m x m o r m n mum a u o f r a imu i i v l e ,

0 ” ( Se e Art . dx

We thu s obtain from (s) o s e ULA TING OIROLE. 135

from which 1 +

A a n a 8 as n to cu a n c c g i , if we reg rd ( ) referri g the os l ti g ir le

2 2 3 as 6 b = 7 ( 0 + (y ) .

dr we shall also have dx

n n all o n o n sin c e r is c o sta t for p i ts the c ircle .

T u o b a n b o fo r cu an d c c am h s we t i , th the rve the ir le the s e 3 2 7 dy d y x r n 1 94 1 an d nc an d — i n c n e p essio ( ) for 3, si e 2 the se o d dx dx dx m mb 1 a at n c n ac am a u e er of ( ) h ve, the poi t of o t t, the s e v l es 3 fo r b o cu a h as k am th rves, it follows th t 2 li ewise the s e 3x

a u n c c n ac . v l e . He e the o t t is of the third order

EX A M P L ES .

F n o f c o n ac cu i d the order t t of the two rves ,

3 2 = x and = 3 x — 3 x l y , y + .

B c mb n n u a n o n x : y o i i g the two eq tio s, the p i t,

o un mm n r is f d to b e c o o to b oth c u ves .

f n a n n u a o n Dif ere ti ti g the two give eq ti s,

y = 3 w y — Gw — 3 , g(U Z(I)

2 t ix

3 ° d y _ 61 F 136 DIFFERE TIA L CAL L N C U Us .

dy n x 1 3 in b cu Whe , , oth rves ;

in both c urves ;

3 d y has n a u in cu . 3 differe t v l es the two rves dx

n a o f n Hen c e the c o t ct is the seco d order.

2 F n c n ac the a ab a 4 x and i d the order of o t t of p r ol y ,

a n x 1 F r. the str ight li e y A ns . irst orde

Find the o rder o f c ontact of = 3 — 2 = x 3 x 27 an d 9 x 28 . 9 y + , y + 3

A ns . S c n e o d order. Fin d the order o f c ontact of

2 y = l o g (x and x

t mm n n 2 ns n a c A . S c the o o poi t ( , e o d order .

2 F n c n ac o f a ab a 4 x 4 and i d the order of o t t the p r ol y , 2 z c c w + 2 3 . n T the ir le y y A s . hird order .

at mu b e a u o f a in o a a ab a Wh st the v l e , rder th t the p r ol

may have c ontac t of the sec ond order with the hyperbola

= - = — x 3 x 1 ? ns . a 1 y A .

Find the order o f c on tac t o f the parabola

2 z x 2 a 2 a z 2 x ( ) + (y ) y,

2 A ns . T . an d the hyperbola xy a . hird order

138 DIFFERENTIAL CA L C UL Us .

1 The envelo e o a seri es O c urves i s tan ent to ever 33 . p f f g y th s curve of e eri es .

Su be an cu o f L N . P ppose , M to y three rves the series is n c n M c n cu L and the i terse tio of with the pre edi g rve , Q its n n n r i tersectio with the fo llowi g c u ve N.

A s cu a ac c nc n c P and u ma the rves ppro h oi ide e, Q will lti tely b e two c n cu o n o f n and o f c u M o se tive p i ts the e velope, the rve Henc e the envelope to uches M S m a m a b e n a n uc an i il rly, it y show th t the e velope to hes y other cu rve of the series .

equa tio n the envelop e of a given series of

Befor e c onsiderin g the gen eral problem let u s tak e the n x m followi g spec ial e a ple . Required the envelope o f the series o f straight lines represented by

= a x ﬁ y + r Cb

a being the variable param

eter . Let the equ ations o f any two of these lines be

= a x m y + r ( b

and ENVEL OP ES . 139

F m 1 and 2 as mu an u ua n can ﬁ n ro ( ) ( ) si lt eo s eq tio s, we d n c n n Su b ac n 1 m the i terse tio of the two li es . tr ti g ( ) fro

c are co - n a o f n n whi h the ordi tes the i tersectio . No w if we s uppose h to approach zero in we have for the ultimate intersection o f c on secutive lines

rn 2 m _ 2’ a a

By eliminating a between these equations we have

2 4 mx y ,

c b n n n n a ua o n o f cu o f whi h, ei g i depe de t of , is the eq ti the lo s

n r c o n o f a n tw o c n cu n a u the i te se ti y o se tive li es th t is, the eq a n u tio of the req ired envelo pe . The u a n an d n ﬁg re shows the str ight li es, the e velope whic h a is p arab o la.

1 n n o w a u n . 3 5 . We will give the ge er l sol tio

Let the given equ ation b e 0 ;

c b a n a am a r n o f whi h, y v ryi g the p r eter , rep ese ts the series cu rves .

To ﬁn d n r c o n o f an cu o f r the i te se ti y two rves the se ies , we c ombine w f ( y y)

x a h = 0 f ( ) y) + ) 1 40 DI FERENTIAL CAL C L F U US.

From (1) and we have

h and n a n it is evide t th t the i tersection may be foun d by co m. binin g (1 ) an d instead of (1 ) and n cu v a ac c n c n Whe the two r es ppro h oi ide ce, h approaches z n Ar a d a b t . 10 m ero, we h ve, y , for the li it o f equ ation

99 a 0 . 4 £1 1 , y, ) ( )

Thus equ ations (1) an d (4) determin e the intersectio n Of n B m n n two c o s ecutive cu rves . y eli i ati g a between ( 1 ) an d (4 ) we shall obtain the equ ation of the lo cu s of these ultimate

n n c ua n n . i tersectio s, whi h is the eq tio of the e velope

A n m c n xam 1 3 6 . pplyi g this ethod to the pre edi g e ple,

= a w ﬁ y + a a

n n 4 A r 1 5 we d ff n a c a and b a t . 3 i ere ti te with refere e to , o t i for ( ) ,

Eliminating a between these equation s gives the equation o f n the e velope,

2 i: 4 x m as b . y , efore

The evo lute o a i ven curve i s the envelo e o i ts 1 3 7 . f g p f

normals .

T n a b u o f A rt . 117 and o his is i dic ted y the ﬁg re , the prop si A r 1 5 n ma be b m t . 3 as tio y proved y the ethod of , follows ' ' h n u n o f n ma at n x T e ge eral eq atio the or l the poi t ( , y ) is b r 2 A t . 99 y ( ) ,

l w

142 DIFFEREN TIA L AL L S C C U U .

Find the envelope o f a series o f c ircles whose centres are o n ax o f an d a o na m m the is X, r dii pr portio l to ( ti es) anc m n 2 2 2 2 their dist e fro the ori i . n = g A s . m x y ( + y ) .

2 Fin d the evolu te o f the parabola y 4 a x according to 1 ak n u a n A rt . 3 7 no ma i n m , t i g the eq tio of the r l the for

— — 3 2 — 3 : 2 = « m x a a m . 2 4 y ( ) A ns . 7 ay (x 2 a )

2 2 F n u o f —= 1 ak n i d the evol te the ellipse , t i the g, 2 g

equation of the n ormal i n the form

2 z b a x tan a b sin S y qt ( ) q ,

where <5 is the eccentric angle .

s 2 2 x § — § A ns . (a ) ( a b )

Find the envelope of the straight lines represented by

x co s 3 d y sin 3 d

n m 0 bei g the variable para eter .

2 ? 2 2 2 ? mn x a . A ns . ( y ) a ( x y ) the le is c te

n n o f ax Fi d the e velope the series of ellipses, whose es

c oincide and whose area is c onstant . The equ ation of the ellipses is

2 2 x y 1 2 ’ a IF

n a ab a am ub c co n a and b bei g v ri le p r eters, s je t to the

2 n a b 76 2 ditio , ( )

2 c alling the c onstant area 7 70

Substituting in (1) the value o f b from (2)

2 2 2 x a y 1 2 7 ’ a 7 0

ff n i n whi ch a is th e only variable p arameter . Di ere i in n c a a t at g (3 ) with refere e to , we h ve EN VEL Op Es . 143

Eliminating a between (3 ) and we have

"‘ 2 4 4 76 x y .

u i o n n a a n b a Seco nd So l t . Differe ti te reg rdi g oth

b as variable .

z 2 x da y db 3 3 61. b

ff n a n 2 a a Di ere ti ti g ( ) lso, we h ve

bda a db 0 .

From (5) an d we have

2 2 x y _ . 2 5 a 5

From (7 ) and

2 ? x y 1 2 2 a b 2

Substituting (8 ) in

z’ 2 4 4 x y 76

F n am r 8 . i d the envelope o f the circles whose di eters a e the 2 double ordinates o f the parabola y 4 a x.

A ns .

Find the envelope o f the straight lines + 1 2 ,

" ” n n a + b k . L Whe L 1 . wn-f-l n + l n A ns . y ye

2 ? x y Fi n n o f i — 1 d the e velope the ell pses 5 , a

l ‘gf l ns . x when a b h. A + y k 144 DIFFERENTIA L OA L OUL Us .

Fin d the en velope of the c ircles passing throu gh the 2 r n c n are o n arab o a 4 a x. o igi , whose e tres the p l y z 3 x 2 d x A ns . ( ) y +

Fin d the en velope o f c ircles des cribed o n the c entral radii

o f an as am u a n o f ellipse di eters, the eq tio the ellipse 2 g “3 d _ 2 2 2 _ 2 2 being + l A 71 3 $13 + y ) — a x " l“ 3/ “2 83

F n n w x c nc and i d the e velope of the ellipses hose a es oi ide, suc h that the distanc e between the extremities o f the n n major a d mi or axes is c on stant an d equal t o k . 2 " s A ua are x k ; An . sq re whose sides ( i y)

146 DIFFERE TIAL A N C L CUL Us .

S nc u c n a n n o a ca x n i e o t i s r di ls, this e pressio for dx

a but o ne a u at an n n un ak h ve v l e y give poi t, less it t es

m a for 8; th t is,

Ga Ga 0 a d

T are c o n n s mu n hese therefore the ditio for a ltiple poi t . If valu es o f x an d y which s atisfy (1 ) also satisfy equation o f the cur v e x 0 f ( 2 y) we have for any su ch poin t dy 0 dx 0

This indeterminate form c an b e evaluated by the method

f A r o t . 53 . The result o f the process of evaluation will be an equation y o f c n c ﬂ u the se o d, or higher, degree with respe t to , th s dx n u n T be determini g several values of that q a tity . his will n m n x m appare t fro a e a ple .

Let xam n mu n mn ca 1 40 . e i e for ltiple poi ts the le is te

2 2 2 2 (90 + u) = 31 )

2 = — x = u ) 0 .

da 2 ? 2 4 x x 2 d x ( y ) , dx

6“ 2 z 4 y(w if) 2 a u ar

Pu n 0 and — tti g , gZ

x = 0 = 0 x = w e ﬁnd , y , or :l: SING ULA R P OIN TS. 147

a u o f x an d x = 0 = 0 a n a Of these v l es y, , y , lo e s tisfy the n cu Let u s ﬁnd equation o f the give rve . the valu e o f

01 —3 n for this poi t . dx

du 3 2 2 2 x 2 x — a x 0 dy dx + y _ = = n x 0 0 . 2 3 2 ’ whe , y dx da 2 x y + 2 y + a y 0 at

A E a ua n b rt . 53 v l ti g y ,

1 2 g 2 6 x 1» 2 y + 4 xy a gx n x = 0 0 whe , y . d

Henc e

The n a ub n an n b n n n origi is do le poi t, the two t ge ts ei g i cli ed ° 45 to X.

A a n ak cu u a n 1 41 . g i , t e the rve whose eq tio is

4 2 3 z u x + 2 a x y d y 0 .

du 3 = 2 2 4 x 4 ax 2 d x 3 a . + y, y dy 148 DIFFERENTIA L CA L C UL US.

du 9 6 Pu n _ 0 an d 1 0 ﬁnd 50 : 0 = 0 t h tti g , , we , y , o be t e dx dy n o n o f c u a n c n n o ly p i t the rve s tisfyi g these o ditio s .

g In n n a u o f ﬁ ﬁ di g the v l es ’ dx

and y2

4 x3 4 O

n x 0 0 . : , whe , y 3 (by 6

E a ua n b A r 5 t . 3 v l ti g y ,

= n x 0 0 . whe , y 6 ayyl 4 d x

E a u a n a a n v l ti g g i ,

Sa y, x = 0 0 n , y . 2 2 whe a — 4 —4 6 y1 + 6 ayy2 a 6 ayl a

3 2 — 2 = 4 Hen c e y1 ( 3/1 ) yb = = 0 o r :t V2. and therefore y1 , y,

n n in u Henc e the o rigin is a triple poi t as sho w the ﬁg re .

150 DIFFERENTIAL OA L L OU US.

1 us s n 43 . C p . W he the branches of the curve are only o n o ne o f n o f cu a n i n ca a cu side the poi t os l tio , th s poi t is lled sp, as P o r , P2.

The c onditions fo r a cu sp are the same as those fo r a point o f cu a n a na c n n o f ma na os l tio , with the dditio l o ditio i gi ry n o f cu o n n e o f n poi ts the rve o side this poi t . Fo r xam ak m cub ca e ple, t e the se i i l parabola ° y = at mg y i )

i exi dx 2

n x 0 W he , dx

There are then two coincident n n B ut nc tange ts at the origi . si e y is imagin ary fo r n egative valu es o f x are no n o n , there poi ts the left n nc n o f the origi . He e the origi is

a cusp .

i n m n n a mu n Co n u a te P o i nts . 1 44 . j g If, deter i i g ltiple poi t,

g n n u o f . are ma n a a a the val es i gi ry, we the h ve poi t of the 3x anc a a an a curve throu gh which no br hes p ss ; th t is , isol ted

n Suc a n i s ca a co n u a te o int. poi t . h poi t lled j g p SING ULA R P OINTS. 151

xam cu e ple, the rve

z' 3 2 d x bx = 0 y + ,

2 3 90 — 2 658 n x = 0 0 , whe , y . dx 2 a y 8

Henc e

dy 6 x 2 b dx y 2 a fl dx = 0 = n x 0 . whe , y Therefore

dv dx

Henc e the origin is a c o n u a o n T a a j g te p i t . his ppe rs directly from the given equa tion _ b ) ’ x from which e v idetltthat b n are esides the origi , there no points o f the cu rve when

‘ x < b.

EX A M P L ES.

Show that the curve

2 2 2 2 4 a y a x x

h as a mu n at n ltiple poi t the origi .

Show that the curve

2 ” l o 1 x y g ( ) ,

h a m n s a ultiple point at the origi . 152 DIFFEREN TIAL L OA OUL US.

Sho w that the c issoid 3 2 x y ’ 2 a — x

h t n as a cu sp a the origi .

Show that the curve

3 2 2 x 2 x 2 x — 5 x — 2 0 + + y y + y ,

h as a c u at n —1 sp the poi t ,

u 5 . Show that the c rve

2 2 2 2 2 2 a m b y ) y ,

h a n u a n at n as c o j g te poi t the origi .

a c u 6 . Show th t the rve

2 2 a x d x b at n y ( ) ( ) , the poi t

h as a c n u a n a b o j g te poi t, if

a o ub n a b d le poi t, if

and cu a b. a sp, if

154 DIFFERENTIA L OAL OUL US.

z dy d y en x 2 x 4 dx d

2 B a x y ( ) ,

S n ua n olvi g this eq tio ,

x = 1 o r 3

To a b ub t u b x 1 n x pply ( ) , we s s it te oth a d d7 — 2 x 4 2 : dx d and ﬁnd n x 1 0 whe , z dx

z d y en x 3 0 . wh z dx

nc n x 1 a max mum He e whe , y is i ; m m m n x 3 a n u . whe , y is i i

Th m x m m f nd m n mum e a u a u o 2 a a u 1 . i v l e y is 4, the i i v l e,

1 In xc na ca ma a n a a u o f 47 . e eptio l ses it y h ppe th t the v l e 2 x n b a mak i ii = 0 a n o f c o n give y ( ) es , so th t either the

i i n i ﬁ T d t o s (b) is sat s e d . his would b e the c ase fo r a poin t o f i n ﬁexio n R o an n , wh se t ge t

r is parallel to OX . He e the ordinate RL is neither a

nl max imum n o r a ml mum .

Bu t there e may b e a maxi mum o r min imum valu e of WW — n en _ _ o Th18 X y, eve wh 2 dx

Th e m o f n in A r 150 . is mo re fully c o sidered t . ethod the followin g article is also applicable to s u ch cases .

Seco n Metho O etermini n Ma xima a nd ﬁf inima . 1 48 . d d f d g

R n u o f A rt . 145 and u n x eferri g to the ﬁg re , s pposi g to MAX IMA A ND MI IMA F OR O E A RIA B LE 155 N N V .

nc a a as a ac P nc a and o n i re se, we see th t we ppro h , y i re ses, ! m . n le av P c a c b A rt . 108 o n g , y de re ses He e, y , is positive gx

and n a o n f T n o P . a the left, eg tive the right, h t is, whe y 3 a max mu m c an m is i , h ges fro to 3x

S m a ma b e n a n as at a i il rly, it y show th t whe , Q, y is m n mum 11 c an m i i , h ges fro to 3x

T a n ma a b e b a n b n c n a tan t hese rel tio s y lso o t i ed y oti i g th t q ,

n which is equal to chan ges Si gn at P a d Q . dx m in Let u s a c n n xa A rt . 146 pply these o ditio s to the e ple , where 2 — gZ= r 1) (r

y d r Here can change sign o nly when x 1 o x 3 . dx

B u n x b e and n y s pposi g to ﬁrst slightly less, the slightly

a an 1 ﬁnd a x — 1 c an m gre ter, th , we th t h ges fro to ( l but n c x 3 n n a a c an m si e is the eg tive, it follows th t h ges fro dx

: I m n x 1 and n a max mu m. n a to whe , de otes i the s e w a ﬁn d a 3! c an m n x 3 an y, we th t h ges fro to whe , d

3, x n a m n mum de otes i i .

5 A a n c n unc n x 4 x g i , o sider the f tio y ( ) (

/ 4 Here a 3 (3 x 2) (x 4 ) (x gl3

3] W hen x c hanges from to 3x

3 n x —2 1 c an m whe , h ges fro to 3x

n x 4 no t c an S n whe , does h ge ig , 2?(23

nc x nn n si e ( ca ot b e egative . 156 DIFFEREN TIA L OA L OUL US .

nc c nc u a a m n m m n He e we o l de th t y is i i u whe x g; a max imum when x 2 ; bu t n either a maximum no r minimum

n x 4 whe . ? d / A s m no t u ab this ethod does req ire i ” it is prefer le to that (Tel o f A r 14 n c n n a n n mu t . 6 c , whe the se o d differe ti tio of y i volves h k wor .

m 1 a se where 00 . i s b e n i c a o 49 . C It to ot ed th t s e dx Ed: f Z times changes sign by passing through inﬁnity instead o ero.

dy 00 Henc e if , dx

f x a u u be xam n as fo r a ﬁn ite value o , this v l e sho ld e i ed, well d y 0 as those gi ven by . dx

Fo r x m u e a ple, s ppose

= —b fB y 0 . (

dy 2 b T n he " d” — fi e e r

henc e we have

00 n x 0 . , whe dx

It is evident that when dy x = 0 c an m , h ges fro dx to indicating a maximum

a u o f c a . v l e y, whi h is Th e ﬁgure shows the max mum na P c i ordi te M ‘ orre

spo nding to a cusp at P .

158 DIFFERENTIA L A C L C UL US.

Henc e the second members of both (1) and (2) mu st be neg ative .

B ak n h u fic n ma r m c an e m y t i g s f ie tly s ll, the fi st ter b ade n u m c a a an s um o f all r n o n eri lly gre ter th the the othe s, i v lvi g 2 3 h h T u n o f m e tc . n r c n m b , , h s the sig the e ti e se o d e er will e a m A n i n 1 b th t of the first ter . s these have difierent sig s ( ) and the second members cannot both b e negative unless

Equations (1) an d (2) then bec o me Z f ( a + h) - (a ) + l§

— —713 0 f ( a h) ( to 1 12 [2

2 The term c ontain ing It no w determin es the sign o f the n m T m b e n a mu a v c m b . a a se o d e ers h t these y eg tive, we st h e

~ f < a > 0 .

If then an d

x mu m f ( a ) is a ma i . S m a ma b e n a i il rly, it y show th t if

and

m n mu m f ( a ) will b e a i i .

If an d similar reason ing will Sho w that fo r a maximum we mu st also

a h ve and and for a minimu m

and

Th e c o n n ma b e n a z as 1 5 1 . ditio s y ge er li ed follows

Suppo se = P < a > 0 o A D MI IMA R E A RIA B LE 159 MA X IMA N N F O ON V .

T n n n a n r a max mu m n o r a he if is eve , f ( ) is eithe i n mum mi i .

n o dd a b e a max mu m o r m n mu m ac c o r n If is , f ( ) will i i i , di g

r o > o.

EX A M P L ES .

2 mum u n x ns . Fi d the ma i val e of a x x . A when x

Fin d the maximum an d minimu m valu es o f

3 2 x n m x m 2 x 9 x l 2 3 . A s . x 1 a a u m 2 gives i , ; x 2 a m n mum 1 gives i i , . Fin d the maximum an d minimu m valu es o f 3 — 2 x 3 x 9 x 5 . A ns . x 1 a m ax mu m 10 + gives i , ; x 3 m n m m a u 22 . gives i i ,

3 2 Sh o w that x 3 x 6 x h as n either a maximum n o r min m i u m valu e .

m n m m 5 . S a a x a u n a x how th t , is i i , whe 2 8

2 2 OL b 2 S o a a a u o f a b . h w th t the le st v l e , 2 z is ( ) s rn d c o s d

Investigate the fo llo wing fu n ctio n s fo r m ax ima o r minima

2 x x n x 4 x 7 6 A s . gives a ma imu m valu e o f y ;

x 10 x 16 a m n mum a u o f gives i i v l e y.

CZ? A m n m A ns . u m n x e i i whe . l o g x

( 66 W a max mu m a u , gives i v l e, 4 a b

‘ 2 10 . 2 tan x t n x s A m x m m a . n a u y A . i when x 160 DIF ERENTIAL OA L O L US F U .

11 . u x 1 n A max mum n c o s x . A s x y si ( ) . i whe

12 . tan x m n mu m n x 3 co t x. A n A y s . i i whe

13 sin x . c o s x a . A y ( ) A ns . max imu m when x

a minimum when x

A ns A m . inimum when a 2 x

4 y (x l ) (x A m A ns . aximum when x a minimum when x 1 n n x either whe 2 . y 2 )5 (2 w 1 r

A ns A m x mu n . a i m whe x a minimum when x n n x either whe 2 .

y = (x

A m n mu n A ns . i i m whe x 5 ; a maximum when x a m n mu m n x —1 i i whe . A y : (2 x a ) (x 2 a A max mum n — m n A ns . x i whe a i imum when x a . 3

PROBLEMS IN MAXIMA AND MINIMA .

10 n uc a a uc o f 1 . Divide i to two s h p rts th t the prod t the squ are o f o ne and the c ube o f the o ther may be the greatest possible . 3 T n —x b e Let x an d 10 x b e the parts . he ) is to a 2 3 max mu m L n u x 10 x ﬁn d i . etti g ( ) , we

t 2 i 5 x 4 x 10 a 0 ( ) ( ) , dx

m x mum n x 4 n from whi ch we ﬁ nd that u is a a i whe . He ce the requ ired parts are 4 and 6 .

162 DIF ERENTIAL OA L O L F U Us .

From the right triangle OPR we have

2 x 2 7 2 y . ( Ct)

The c onvex surface o f the cylinder

- 2 2 wx 2 y — x

This will b e a maximu m when z z 4 u r x x a max mum is i . This is found to b e when x V2 the radius o f the base o f the required c n yli der . i» r F m i nc a u f ro th s, y He e the ltit de o the cylinder is V2 / r \ 2 .

A n other solution of the problem is the following

S nc c n x u ac 4 wx ut i t x b e a m x i e the o ve s rf e is y, p y, to a imum. d a dy 00 0 3! . (5) dx da:

B ut m a x o. fro ( ) , (c) 58 l E m n a n — m b and c a x = li i ti g fro ( ) ( ) we h ve y, which ili c mb n a am u as b o i ed with ( ) , gives the s e res lt efore .

A c an a f n 5 . re t gul r pie ce o pasteboard 30 i ches lon g and 14 inches wide has a squ are cut o ut at e ac h c orner ; ﬁnd the side o f this squ are so that the remainder may form a b o x o f maxi

m n n . mu c A ns . 3 nc o te ts i hes .

n u a u h 6 . Divide a i to two parts s ch th t the prod ct of the mt power o f o ne and the nth power of the other may be a maxi n d n m ns Th u a are na m a . u . e m A . req ired p rts proportio l to

A n b n in a b a 3 m m n a n 7 . perso ei g o t iles fro the e rest poi t o f b ac ac i n m a ac 5 m the e h, wishes to re h the shortest ti e pl e iles MAX IMA A ND MINIMA F OR ONE A RIAB LE 16 V . 3 from that point along the shore su pposing he can walk 5 miles

ut ro w n at a o f 4 m an u u an hour , b o ly the r te iles ho r , req ired

an the plac e he must l d . n m A ns . O e mile fro the place to b e re ached .

f a a c u a n a a u 1 1 8 . The to p o pedest l whi h s st i s st t e feet high ’ is 25 feet ab ove the level o f a man s eye ; ﬁnd his horiz ontal distanc e from the b ase o f the pedestal when he sees the statue

a n n . 3 a . A s 0 f s ubtending the gre test gle eet .

b an u ax . T u h a n a d c a a 9 hro g poi t ( , ) , referre to re t g l r es ,

a n b e d a n m n ax a an str ight li e is to r w , for i g with the es tri gle o f the lea st are a . Show that its interc epts o n the axes are 2 a

and 2 b .

10 . T u n a b a n d a n uc a hro gh the poi t ( , ) li e is r w s h th t the ax n m part interc epted between the es is a mi i um . Showthat its ll % I length is (a b )

n an a o f a n ﬁn a 11 . Give the sl t height right c o e d its ltitude when the vo lume 18 a max imum . a A ns V3 2 = 12 . G n a n o n ax o f a ab a 4 a x at ive poi t the is the p r ol y ,

~ h m h x n the distance . fro t e verte ; ﬁ d the ab s ciss a o f the point

u n a . A ns . x h o f the c rve e rest to it 2 a .

F n max mum n a an 13 . i d the i recta gle th t c b e ins cribed in an - ellipse whose semi ax es are a and b . / / A ns . The are a 2 and b 2 a a 2 sides \ \ ; the re , a b .

14 . A c an u a b o x n at to a ua b a re t g l r , Ope the p , with sq re se , is to b e c o nstruc ted to c o ntain 10 8 c ub ic inches . What mus t b e its dime nsion s to require the lea st material ?

A ns . A u n f b a 6 n 3 c o . ltit de , i hes ; side se , i ches

F 15 . ind the altit ude o f the right cylinder o f greatest vo lume n i s cribed in a sphere whose radius is r . 2 r An3 . 164 DIFFEREN TIA L OA L OUL Us

F n 16 . i d the altitude o f the right cylinder ins cribed m a

S a u r n n u ac a max phere whose r di s is , whe its e tire s rf e is imum .

A ns .

17 . Find the altitude o f the right c one o f gre atest vo lume

n c b in a a u r . n i s ri ed sphere whose r di s is A s . {f r

u f 18 . Find the altit de o the right c one o f max imum entire in surfac e inscribed a sphere whose radius is r . 2 " ° A ns . 23 ( 15

Find the altitude o f the right c one o f le ast volume c ir c ums crib ed ab out a sphere who se radius is r

' 4 r nd um A ns . a ud a f Its ltit e is , its vol e is twic e that o the sphere .

F n a u o f a s an e cir 20 . i d the ltit de the le st isos c ele tri gl o m -ax s are a and b cums crib ed ab out an ellipse wh se se i e , the b ase o f the triangle being parallel to the major axis . 3 A ns . b .

- A an n a n to m ax are a 21 . t ge t is dr w the ellipse whose se i es

s a a n c b ax a m n mum. and b , uch th t the p rt i ter epted y the es is i i

Sho w that i ts length is a b .

h c n o f a a a f d 22 . T e lower or er le f , whose width is , is ol ed over so as j ust to re ach the inner edg e o f the page . Find the n n o f c a width o f the part folded over , whe the le gth the re se is

A ns . a . a minimum . 4

n c d n x am ﬁnd n a a o f 23 . I the pre e i g e ple , whe the re the tri

mum . angle folded o ver is a mini

n a . A ns . Whe the width folded is g

166 DIFFERENTIAL A O L OUL US.

A s h and It are n n n o f ac i depe de t e h other, this is equivalent to

du du 0 and (2) dx By

Equation (1) then bec omes

— 2 ? f(w k f x A7. 2 B kk Ck ) ( , y) %( )

z z z d u d u d u where ’ B z ’ z ar ar ay dy

2 2 A " 3 7“ A OL -3 70 2 2 = + But Ah + 2 Bh7t + OIr . (3 )

In order that (3 ) may preserve the s ame Sign fo r all small 2 a u o f h an d Is n c a a v l es , it is e ess ry th t A C B should b e n a num a o o f 3 positive ; for if eg tive, the er t r ( ) will be positive

n k : 0 and n a n A h B k whe , eg tive whe 0 . Henc e we have as an a na c n n a m ax mum dditio l o ditio for i ,

B 2 4 A O. ( )

The sign of (3 ) then depends u pon that o f the denominator n A . He c e for a maximum we mu st have

A < O.

S m a ma be n a fo r a m n mum a u o f u i il rly it y show th t i i v l e , w e mu st have (2 ) and together with

A > O.

It may b e n otic ed th at (4) requires th at A an d C should A b a a am n . nc C e . h ve the s e sig He e if is positive, will lso The xc na c a e eptio l ses, where B 2 =A C, o r where A = O B = 0 = 0 , , C

u n n a no t c n m req ire further i vestigatio . We sh ll o sider the here . MAX IMA AND MINIMA F OR TW O VA RIA B LES. 167

1 The c ndi n a max mum o r m n mum a u o f 54 . o tio s for i i i v l e u x ma b e a as f ( , y) , y rest ted follows Fo r a max mum o r m n mum either i i i ,

0 nd , a

Fo r a max mum 0 and i , ,

Fo r a m n mum 0 and i i , ,

n Th r i l s A m n Functio s O ree Va a b e . a a n 1 55 . f si il r i vestig tio 1 n a in A rt. 53 as c n o f a max mum o r to th t , gives the o ditio s i n m m v ulue u = x z mi i u of f( , y, ) m x mum o r m n mum Fo r either a a i i i ,

Ga Ba Ba

dx dy

z 2 i z d u d n d u

2 ff dxdy 8x dy

Fo r a max mum 0 and A 0 i ,

a m n mu m 0 and A O for i i , > , > ;

2 2 d v d a d a i , 2 6x dxdy dxdz

2 Q z d a d u d u A ’ where z dxdy dy dydz

2 z 2 d a d u 6 u ’ dxdz" dydz 168 DI FERENTIA L OA L F OUL Us .

EX A M P L ES.

F n m 1 . i d aximum value o f

u 3 ax x3 ’ y y .

du da 3 ax dx dy

2 ’ 6 a d u AlSO 2 z (9x dy

n 1 A 4 A rt . 15 a pplyi g ( ) , we h ve

2 2 a — x = 0 and d x — = y , y 0 ;

whenc e o r

Th e a u x 0 0 v l es , y , give

z i ( n d u d n i 0 2 z dx dy dxdy

2 A r no a 1 4 . which do t s tisfy ( ) t . 5 nc n o t a max mum o r m n mum He e they do give i i i . The a u x a a v l es , y , give

’ 2 z d a d a d u

2 2 6x dy dxdy

2 n Ar 154 . which s atisf both a d t. y . ( ) 3 n m ximum a u f He c e they give a a v l e o i t which is a .

F n m x mum a u o f x z ub c c n n 2 . a i d the i v l e y , s je t to the o ditio

2 2 2 x y z + + 1 . 2 2 2 a b c

2 2 2 (13 F m 1 E- ro , , 2 ; i a li

"’ 2 2 an d as xyz is numerically a maximum when x y z is a max mum ut i , we p

170 DIFFERENTIA L OA L OUL US.

F n i d the values of x and y that ren der

s in x i S n y + co s (x + y)

a max mum o r m n m i i i u m.

A ns . A ml nl mum , when x

’ a max mum n = i , whe x y

7 . Find the maximum valu e o f

Let a b e x an d a x — the p rts , y, y. — T n i t x a x b e a max mu he y ( y) , to i m.

6 l a i — — 2 2 a 2 x = 0 ax — x —2 x = y y y , y 0 . da; (I;

These equations give x y 3.

n He c e a is divided into equ al parts .

N TE — n m na u o f b m O Whe , fro the t re the pro le , it is n a a max mu m o r m n mu m evide t th t there is i i i , it is oft en unn ec ess ary to c onsider the sec ond differential c ﬁ n oef c ie ts .

m n a n x z uc a x zp ma Divide i to three parts, , y, , s h th t y y be m m m a axi u . x y z m n p m + n + 10 MA X IMA A ND MINIMA F OR TW O VA RIAB LES. 171

/ 1 3 0 n o u ar such that co n nu o d . Divide i t fo r p ts the ti ed pr

n t o f S u ar o f c o n cu b o the ﬁrst, the q e the se d, the e

r and ur o u ma the thi d, the fo th power of the f rth, y m m b e a max u . i 1 ns . 6 9 2 A 3 , ,

3 Given the vo lume a o f a rectangular parallelo piped ; ﬁn d

n m n mu m whe the surfac e is a i i .

n b ns a a a cu . A . Whe the p r llelopiped is e

An open vessel is to be c on stru cted i n the form o f a

c an u a a a c a ab o f c n a n n 108 re t g l r p r llelopiped, p le o t i i g c ub c n a mu b e m n n i i ches of water . Wh t st its di e sio s to require the least material in c on struc tion

L n n in in n d w . 3 A s . a 6 . e gth idth, height,

F n c o- na o f a n sum o f ua i d the ordi tes poi t, the the sq res o f d anc m n n whose ist es fro three give poi ts,

50 55 x ( 1; ( 2; ( 3,

m l m m 1° m u ' t A ns . w + w + + ge n . t ) . é

the c entre o f gravity o f the triangle join ing the given n poi ts .

F n um o f a an 5. i d the vol e the gre test rect gular parallelopiped that can be in s cribed i n the ellipsoid

e 2 i x t

- l =n 4 “ 1 . 2 2 e (1 5 ‘i

174 G CAL C UL Us INTE RAL .

o f a kn n unc o n o r i n u c n to a m uc ow f ti , red i g it for where s h n o n o b All un n f n co . c n a b e a re g iti is p ssi le f tio s c di fere ti ted, b ut all c ann o b e n ra a n a c anno t i teg ted ; th t is , their i tegr ls t x i n m k n wn un n b e e pressed ter s of o f ctio s .

i n i l s Elementa r P r c e . 2 . y p

a n a a ( ) . It is evide t th t we h ve

2 2 2 x dw = x 2 o r x dx = x — 5 + , ,

2 as well as 2 x dw= x ;

? 2 nc at 2 an d x 5 are u nc n ac o f diﬁ eren si e + f tio s, e h whose

ti l Z a } a s is ch .

2 In n a 2 so a: x c ge er l d ,

where 0 den otes an arbitrary c onstant c alled the co ns tant of

integrati o n . n a in m n a m n c u m Every i tegr l its ost ge er l for i l des this ter ,

e a m t c n an o f n a o n i n w . We sh ll o i this o st t i tegr ti the follo

ln n a as c an a b e a n n c a . g i tegr ls, it re dily dded whe e ess ry

S nc d u v (b) . i e ( :l:

it follows that

(da j : dv j : dw)

That is we integrate a po lyn omial by integrating

a m and a n n n . r te ter s, ret i i g the sig s

a u a du Sin c e d ( ) ,

a it follows that a d .

T a a c n an ac ma b e an m o ne h t is, o st t f tor y tr sferred fro side

m u aff c n n a . o f the sy bol to the other, witho t e ti g the i tegr l Y IN TEGRA L S 175 ELEMENTAR .

Fun a menta l Inte ra ls . a no w a o f 3 . d g We sh ll give list

o mu a c ma be a as un am n a and to f r l e, whi h y reg rded f d e t l,

r whic h all integ als mu st ultimately b e redu c ed . We shall then c onsider i n this c hapter s uch e xamples as are integrable b mu a c o r a m m an y these for l e, either dire tly, fter so e si ple tr s ma n for tio .

u s in os u d a .

in u da co s a .

ec u du tan a .

g V . o sec u d u c o t a III .

u t n e c a u du s e e u .

c a co t a d a c c u ose ose .

an u a l d o g s e c u .

o u u l i t d o g s n a .

l XIII . e c u da o g ( s e c u tan a ) l o g tan

“ o s ec u du lo g (co s e c u co t u ) lo g tan g 176 I CA L C UL Us NTEGRAL .

1 __ a u 10 g 2 a a u

V . X II c o s

V X III .

1111 1. cosec

V GI' S ? V2 a u a”

o . a nd f I II .

To I derive . ,

nc si e n 1 M eta ( ) , therefo re

n “ a n 1 M eta ” ( ) (n 1 u du b 0 A rt . 2 ) , y ( ) .

u --l n i Hen c e n 1

F m or ula follows dir ec tly from

du at l o -— g a . u ’

b e n c a It is to oti ed th t I . applies to all valu es o f n exc ept n 1 . Fo r a u this v l e, it gives

o n

w e 0

Fo mu a . fo r a n c a o f r l II provides this f ili g se I .

17 8 IN TEGRA L C C U Us AL L .

2 6 “ lo “ g x . ac 2

2 4 “ x ‘ 3 ( 3 6 2 % 1 9 n 11 . ( a 2 x 5 31 i

5 2 x7 12 Vi m; ﬁ x . 2 5

90 2 ( 4 a: 1 da; 13 . ( ) 3

n a a a x an n x are I tegr te lso, fter e p di g ( + How results rec onciled

2 m” 4 a " a ” lo w 53 ) g .

2 x 15 . (

2 n a a a mu n x 2 41: 2 b a: 1 an I tegr te lso, fter ltiplyi g y , d m c o pare the two results .

2 s ? 2 2 3 3 a n: a % 2 am at da: a x m 16 . ( ) ( ) 63 ) { 5

1 8 —-lo ac x a 3 b . 3 g ( ) a m + 3 bx 3

n a a a mu n num a an d n m na I tegr te lso, fter ltiplyi g er tor de o i tor

b 2 an d m a tw o u . y , c o p re the res lts

- clx na 2 18. ( y Y INTEG ALS 1 9 ELEMENTAR R . 7

a: log (2 a:

2 3 90 m

t « a .

f (i )?

doc : %[lo g (9c

2 g , 3 ( a

3 3 % (x 1) (m 5) dx gar

l l ; ‘ ‘ 3 3 l 3 l 7f u i n. 1 x S ggest o (x ) ( (a 5 w ﬁ(£1: x ) .

n —l

n n (x + 7L) n (o r n )“ ml M n m u estio n . u u a and n m na b co S gg ltiply er tor de o i tor y .

m I l‘ The n n a a be a ua b . a mu followi g i tegr ls y ev l ted y , fter 2 tiplying the bin omial under the radical sign by a:

d” - - - e 2 3 ~ 29 . i G m 1) x da.

2 2 “ 3 2 a a: dw)

2 2 %L -2 (a x 1 ) 1 80 I C C S NTEGRAL AL UL U .

T ma e n m x n his y b obtai ed fro E . 33 by substituti g w a x for .

P o o I r n b a n d . T a e r o . a n 5 . f f III V hese evide tly o t i ed

r f n n di ectly from the c orresponding formulae o differe tiatio .

EX A M P L E8

III FO R ORMU LJ E . A ND IV F .

. n l o g a m l o g b

8 2 I CAL CUL s 1 NTEGRAL E.

B T n m y rigo o etry, f“f 2 c osec u co t u 7" i t 2 in s s 5co

If w e ub u in 35 u u s stit te this + for , 2

Ow “ 1 I . sec u tan u tan 0 0 "M

b a n c n m o f and o t i the se o d for s XIII . XIV.

EXA M P L ES — FOR FO R MU LA; V . XIV.

o s x da; sin 2 m c o s 2 m (sin 2 x c 2 ) %( ) .

° = g sin 3 w) da

— _4 " 0 0 8 in (a + bw) c os (a b

doc 1

2 c o s 3 w

— = c + tan dx 2 an g+ s e c

w t a x c osec a c o ) .

2 n 417 (tan x co t m) dw ta a c o t .

2 (se c s: tan w) d:c 2 (tan a: see m)

sin cc ale: 1 —- x lo g (a b co s ) . b

2 z m b sin m — 2 (b a ) Y I TEG AL ELEMENTAR N R s . 183

— 2 (tan 2 w 1) da: tan 2 w + l o g c0 8 2 ca

c o sec a: 1 co t x 1 dw —x c c w—l o 1 co s a ( ) ( + ) ose g ( + ) .

2 n a a: 2 10 tan x (s e c at: c osec w) dw ta co t g .

x- z 93 n in in wdoc s 2 x. u 2 s

s — z x daz= g s in 2 x s m . p 2 p

da' 2 (sec a: tan x)

dx él o g tan — _ _l o g 1 tan a tan a an actan (w+ a ) dx w t an a

ec wsec (cv + a ) dw

— P ro o O XV. . 7 . f f XX T o XV. derive ,

To I derive XVI , 184 I CAL C UL Us NTEGRAL .

To derive XIX.

To . derive XX ,

da vers 2 2 a u u u

‘ 4 1 17 a 2

6 “ 1 1 “ l it is evident that d c ot a

nc x o n ma be u as n a i n X He e either e pressi y sed the i tegr l V. am w a n n m f n In the s e y we obtai the sec o d for s o XVII . a d

XIX. h mu I n r n in a d . a e T e for lae XV . XVIII i serted the list o f m n n a b cau are a m XV. a d . i tegr ls, e se they of si il r for to XVII ,

c n n . respe tively, with differe t sig s I To XV . derive , 1

2 a 2 a henc e

2 a 2 a 1 1 — - — [lo g (u a ) l o g 2 a 2 a

Or we may integrate thu s

186 I C CU Us NTEGRAL AL L .

1 — vers b 1 b

1 ‘ 1

v e rs 3 cc. V8

1 bx a 1 bw+ a l o g ° 2 a b bw + a 2 a b bat — a

5 —1 3 ” V 6

1 5 w /3 — 2 — \ — dx = log (3 13 2 2 3 a:

Th e s ame formulae may be applied to expression s involving 2 a x b o r m a x b b c m n ua , y o pleti g the sq re m n n n w T u c a . with the ter s o t i i g h s,

dx _1_ 2

— 1 2 513 1

1 V2 _ 1 x 5 10 g . 4 513 — 1 TA R Y I TEGRAL s 1 87 ELEMEN N .

doc — 3 — \/1 + 3 w w VI73

— log (a: 2 4 cv

" 1 t n s e c a tan (msee a a a ) .

1 3 a> + 1 3 4 a + 3

“ 2 a m+ b tan C P E I HA T R I.

INTEGR A TION O F RA TIO NA L F R A C TI N O S .

P r limi na r ra ti n e O e o . 8 . y p If the degree of the nu merator ua o r a an a o f n m n a is eq l to, gre ter th , th t the de o i tor, the frac

n u b e u c to a m x uan b v n tio sho ld red ed i ed q tity, y di idi g the nu m a b n m n er tor y the de o i ator . Fo r xam e ple, E — 2 E m3 + 1

5 — 4 2 cv 3 x + 1 — 2 a: —3 4 2 4 2 56 4-18 m + x

The degree o f the numerator o f this new fraction will be an a o f n m n Suc n nl less th th t the de o i ator . h fractio s o y will be c onsidered i n the following articles

Facto rs O the eno mi nato r . A a na ac n n 9 . f D r tio l fr tio is i te a b c m n n a a ac n deno mi gr ted y de o posi g it i to p rti l fr tio s, whose f n n m n nato rs are the factors o the origi al de o i ator .

o w n b T r E ua n a a N it is show y the heo y of q tio s , th t poly n m a o f nth ct as ma be o i l the degree with respe to , y resolved n ac o f into f tors the ﬁrst degree,

- a; a a: a m a a; a . ( 1) ( z) ( 3) ( n )

are a o r ma na but ma na fac These factors re l i gi ry, the i gi ry u in a o f m tors will o cc r p irs , the for

—1 and m— a — b / — 1 ao , \ , whose produ ct is ( a a real factor o f the sec ond degree .

1 90 CA L C UL s INTEGRAL U .

2 x + 6 w —8 2 1 2 nc He e 3 x — 4 a: a: w — 2 ﬁzz

dx 2 log a lo g (x 2) 2 10 g (x + 2)

° 2 ( 50 2)

A m o f n n A B O n shorter ethod ﬁ di g , , , is the followi g

in 2 m= 0 B an d d a a If ( ) we let , C will is ppe r from the u a n and a a eq tio , we sh ll h ve

8 —4 r = A o A 2 . ,

‘ = S m a 0 1 B 1 . i il rly, If

If a: OI‘

EX A M P L ES . — 3 x 1 2 dx l o g [(ac 3 ) (a: 2)

dcv = l o g w _ w3

2 — 2 w + 2 w c o s a + OOS a 2 2 x + 2 x + sin a

2 {Add 513 1 (1) — 1 16 + - a: 2 3 log ( (m2 2 6 (90 + )

ce dar 2 x —4 x + 1

2 ' l 2 f —2 l o g x , ( 2 V3

x — 2 — ﬁ z 1 V 10 g (iz w RA TIONA L FRA CTIONS. 191

3 4 — — 5 x + x 8 2 2 4 50 + 10 g 3 — 3 (13 4 x (x + 2 )

5 — 3 90 5 az + 4 m (a:

A E t rs O th mina to r a ll O he r 1 C S . Fa c o e eno t i st e ree 1 . II f d f f d g ,

s e e e te a nd o m r p a d .

m o f m n f u m Here the ethod de c o positio o Case I . req ires odi

ﬁcatio n . Su fo r xam a ppose, e ple, we h ve

w( az

m o o f r c di n c a u If we follow the eth d the p e e g se, we sho ld write 3 90 4- 1 A B D — — — o x (x 90 513 1 w l w l

B ut sinc e the c ommon deno min ator o f the fraction s in the sec ond member o f this equatio n is (n(a: their su m c anno t be equ al t o the given fractio n with the denominator n( a: To m b c n w e a um eet this o je tio , ss e

3 90 4- 1 A B w(w a; ( 115 (93

a n o f ac n Cle ri g fr tio s,

3 90 + 1 = A (w 1 ) + Dw(m

— ’ — 2 3 A + O 2 D) an

SA B x — + ( + C + D) A .

nc A D = 1 He e + , —3 A — 2 D = O + O ,

SA + B —O + D = O

A = 1 .

nc —1 B = 2 D = 2 W he e , , . 192 INTEGRA L CA L C UL s U .

2 1 +

— 1 l o x + 2 l o x —1 g 2 g ( ) -m — ( sr l ) w l a: (ax l 0 g ( cv a:

M P L EX A ES .

3 — ‘2 — 93 4 a + 4 x 513 2

2 — 3 113 2 _ 12 w + 19 da: 3 1 2

2 x

- — 2 1 ( ax F1 )

5 — — 2 x 5 93 3 x 2 x + 3 — 2 l 1 2

d x x 1 10

— 9 ( 2 03 5 2 x + 1 l — - (m x ( cc 1

6 — — (8 00 1 ) dx _ 12 x + 1 108 w 61 7 . x + 24 lo g x ‘2 — 3 2 (2 a x) 2 x 4 (2 w 45 o g wc z

no mina to r co nta ini n acto rs O the s eco n 2 CA SE . e d 1 . III D g f f

e ree but n o ne r e ea te . d g , p d The fo rm o f dec omposition will appear from the following

xam e ple, 5 a: + 12 da:

194 INTEGRAL CA L C UL US — — 3 x 2 ) da: 3 x 3 ) dx dx g g + 2 — 2 — 2 w 2 w + 5 x 2 w + 5 x — 2 w+ 5

— 2 (cc 1 ) (x - 2 ao + 5 )

EXA M P L ES.

4 d“; 1 CD ‘ l l tan w 4 2

2 w dx 1 1 ( a: I f + 10 g . — 2 (w 2 (w 1 ) 4 x + 1

3 — (00 6 ) dx “ (13 te n 4 2 w + 6 w + 8 we 2 2 2 V2 V2 + .

-- _. l o g t tan 2 2 Vg V3

W )

4 2 m + 6 x + 25

‘ 1 mL — m“ 1 " 1 lo g an « rtan RA TIONA L FRA CTIONS. 195

4 da: 1

4 - 1 2 — 50 4 w oox/2 + 1

2 x 0 0 S 2 a + 1 4 2 w + 2 w 0 0 8 2 a + 1

2 n w 2 wsi n a 1 C 2 a' si a + + os a. _1 cos a l + tan 2 — — 2 4 n 2 x si n a + 1 2 1 93

‘ min to r nt in in CA SE IV. eno a co a a cto rs O the seco nd 1 3 . D g f f h h e e ree s o me O w ic a re r ea te . d g , f p d

T a a am a n a b . c C a a . his se e rs the s e rel tio to se III , th t C se II n u m m n f b a a I. a d a a ca o e rs to C se , req ires si il r odiﬁ tio the n partial fractio s . Fo r illustration take 2 ﬁ + x + 3 (w? W e assume 3 2 x + w + 3 Am+ B 0 cc + D + 2 2 2 (x + (90 + 93 + 1

-B

= — = = 1 B 3 O. A , , D

3 x + m+ 3 —a: + 3 2 513 T f here ore 2 2 (x + 1 )

a dm dx = (M 1 ) (eon

2 (90 4

o n a ac n u se n m T i tegrate the l st fr tio , we the followi g for ula o f uc n red tio , 1 96 INTEGRA L CAL C UL US.

T mu a b e i n C a I . bu his for l will derived h pter V , t the

u n c an no w b f r n a n b m mb r st de t verify it y dif e e ti ti g oth e e s . It enables u s to integrate the expression peculiar to

this case b mak ing it de y 2 2 (2 + a ) By suc cessive applications the given integral is made to depen d w ‘ l u ma u n c l tan lti tely po 2 whi h is a: a a a

To a mu a mak a =1 n z pply this for l t we e a d n 2 .

We then have

die a: 1 _1 + -tan a: 2 2 (a' + 2 2 63 + 1 ) 2

1 3 x ‘ l nc an cc whe e ,

3 w 1 3 + ‘ 1 2 —tan 90 o x 1 + + l g ( . z

A s another example in the integration o f a partial fraction

i n Ca IV. c n se , o sider

3 a + 2 do: 2 2 (az ( 113

— (2 x 3 ) dw 3 2 — 2 (x 3 w + 3 )

doc 2 — 2 (x 3 x + 3 ) where z a:

198 INTEGRAL GALGUL Us .

3 x + 8 x + 21 2 dz log (a: 4 33 + 9) (w2

4 2 (33 + 56 + CHAPTER III.

N B Y R IZ I N NTEGR A TI INTEGR A TIO A TIO NA L A T O . I ON

B Y S U B S TITU TIO N.

1 4 A s c n c a fo r n a n . the pre edi g h pter provides the i tegr tio f a na ac n a an a na a b a o r tio l fr tio s, it follows th t y r tio l lge r ic n n functio is i tegrable . Some irrational expressions may be integrated by substitut i n a new a ab a o ld a ne w x g v ri le, so rel ted to the , th t the e pres r n sion shall be atio al .

Ex ressio ns i nvo lvin o nl ra ctio nal o wers O w. Suc 1 5 . p g y f p f h m ma be a na z b a um n a: z” n for s y r tio li ed y ss i g , where is the least c ommon multiple o f the denominators o f the s everal n x n n fractio al e po e ts .

dcv Tak xam e for e ple, l i x + aﬁ

“ 5 £6 : z dx = z d , 6 z ;

ll 3 2 a Z % , £13 2

5 6 z dz

3 2 z z

3 2 72 2

f Sub u n in p a stit ti g this, z , we h ve

da: I i i‘ 2 x —3 x + 6 m 200 INTEGRAL GAL L OU US.

Ex ressi o ns invo lvin o nl r ti na l 1 6 . a c o o wers O a bx p g y f p f ( ) , a be a na z b m f n a c m y r tio li ed y the ethod o the precedi g rti le .

da) Tak xam e for e ple, s f (x (x 2)

“ 5 A u m x 2 z dx . ss e , 6 a dz

‘5 dx 6 z dz z dz 5 4 z z 1 (x z + + 6 [z l o g (z

Substituting 2 we have

dx 6 (x 6 lo g [ (x w_W' 6 w

EXA M P L ES.

i" x dx 4 it lo g (x 3 3

1 2 lo g x 24 l o g (xT7

2 6 x + 6 x + 1

i g_(x 3 (x i )

202 INTEGRA L OA L G L S U U .

—1)

a u m as in c n a If we ss e, the pre edi g rticles,

x n x and c n u n a o f dx in m ' the e pressio for , o seq e tly th t , ter s o f m z n a ca . To b c n um , will i volve r di ls eet this o je tio we ass e

are expressed rationally terms o f z.

ak xam T e for e ple,

2 A m z —x —x 2 = z — 2 zx ssu e , + ,

(2 x

ﬁ —z + 2

2 2 — 1

2 dx 1 z — 2 z INTEGRA TION B Y RA TIONALIZA TION. 203

2 Ex ressi o ns co ntaini n x a x b . 19 . p g 2 n z in c a n c a b ax —x To ratio ali e this se, it is e ess ry to resolve + T a b e a un n ac . c i to two f tors hese f tors will re l, less the given 2 n f F r radical V b a x x is imagi ary for all values o x. o

2 2 a’ b + ax —x = — + b

2 — ( V a + 4 b a ) +

2 T ac are a un a 4 b n a but n hese f tors re l less + is eg tive, the 2 b a x — x n a all a u o f x and c n u n + is eg tive for v l es , o seq e tly 2 x m V b a x is i aginary . R n ac u eprese t the two f tors th s,

— 2 = b + ax x (a

No w assume

T u x x a na in m o f z h s is e pressed r tio lly ter s .

Take

— Assume V 2 + x x)

2 — 2 2 z 1 l + x = 2 —x z x ( ) , 2 ’ z + 1

3 x

T herefore, INTEGRAL A 0 L OUL US .

Substituting 2:

EX A M P L ES .

_ ‘ 1 2 2 tan (x x + 2 x

8 + 6 x

x —1

2 Inte rati o n b Substi tuti o n . T m 18 u d fo r O 7 0 . g y his ethod se

a na za n as n in c n a c but in r tio li tio , show the pre edi g rti les, other c ases the introduction o f a new variable often simpliﬁes

C APTE H R IV .

R NTEG A TI N Y . I O B P A R TS . INTEGR A TION B Y S U C . C ES S IV E RED N U C TIO .

2 Inte ra ti o n b rts F m P a . u n 1 . g y ro the eq atio

d uv u dv v du ( ) , we o btain b n a n b m mb , y i tegr ti g oth e ers, .

u dv da .

Henc e dv uv

The use o f 1 ca i nte rati o n b arts ( ) is lled g y p . Let us a xam pply it, for e ple, to

lo x g dx.

Let u lo x n dv x dx g , the ;

dx nc nd 7) whe e , a x 2

Substituting in we have

2 - -" lo g x x dx lo g x g

2 f £ lo g x 2

The u n u a u n c ac u st de t sho ld c ref lly oti e how the f tors , v du ccu i n c as b e ab a u , , o r the pro ess, so to le to pply it witho t n x such a formal substitution as in the pre c edi g e ample . n n ua n a a c n O referri g to the eq tio we see th t, fter sele ti g u a c a n n n a as lo x b a n for ert i factor of the give i tegr l, g , we o t i ﬁ m in n m m b n a n as the rst ter the seco d e ber, y i tegr ti g if this INTEGRA TION R Y PA R TS. 207 factor were con stant ; also that the expression followin g the

n am as c n m ac seco d is the s e the pre edi g ter , with the f tor

x f n lo g replaced by its di fere tial . Take fo r another example

co s x dx .

A u m n u c o s x ﬁnd ss i g , we 2 x co s x dx = co s x — — 1n x S dx) . 2

But as the new integral is no simpler than the given we a n n n b a ca n o f g i othi g y this ppli tio the proc ess . If u x ﬁn , however, we let , we d

xco s x dx x sin x sin x dx

x in x s co s x.

0 7 .

EX A M P L ES.

lo g x dx o g x

‘ l lo g x dx = o g x

sin x d —x i x co s x + s n x.

2 lo g (x 2) dx = (x 4 )lo g X/x + 2

ax eu dx

2 x 1 c “ 1 + ‘ 1 tan x dx tan x —£ 2

‘ l “ 1 2 in x dx x sin x x . 208 INTEGRA L OAL O L s U U .

_ - 1(P 1 + v v —i ﬂ m 10 . 2 1 1 ( 3.

x 1o x l o l 2 g g (H ) (w + 1 ) 90 +

2 " 1 x s1n x dx

In each o f the following examples integration by parts must e a u b pplied s cc essively.

f z 2 x e dx = x 14 . (

2 “ w 3 x 6 x 6 e ﬁ e dx + 2 a a a

3 2 2 — x (lo g x) dx (lo g x) %lo g x

Eg m— 2 17 . g x

m Fo rmulae O R e uctio n. T are ul a b h c 2 2 . f d hese for e y w i h n a the i tegr l, “ " x a bx d ( ) , may b e made to depend upon a similar integral with either m m o r n um ca m n . T are u uc u a p eri lly di i ished here fo r s h for l e, as follows,

bx" ) 1’dx L“ w M a + bx d A

210 INTEGRAL GAL OUL US.

Sub u n 2 i n and an n a stit ti g ( ) tr sposi g, we h ve

(np + m 1 ) b

m n 1 a a p ( ) z dx.

n b n m 1 b a A Dividi g y ( p ) , we h ve ( ) . If in (3 ) we substitute

m and an a tr spose, we h ve

' m' (m 1 ) a x zp dx

m' 1 1 ' x + zp + (np m n 1) b

Om n acc n and n b we a itti g the e ts, dividi g y h ve

Deri vatio n O Fo rmulae B a nd . n a 24 . f ( ) (D) If we i tegr te

m b a x zp dx ca n u zP w e a y p rts , lli g , h ve

m“ “ " b - - ? p +l P l r 1 dx z z a dx. m 1 m 1

m 1 (m 1) x + zp n

m ” m ‘ 1 x zp dx (a bx ) x zp dx

‘ l a dx b

Eliminating from (1) and we have

m l m p m+1 p P x (np m 1) x z dx x z np a x z d .

n n m 1 a B . Dividi g by p , we h ve ( ) ED UCTION 21 FORM ULAE OE R . 1

If in (3) we substitute p

nd an a a tr spose, we h ve

' m p ' “ x z dx dx.

m n acc n and n b n 1 a a O itti g the e ts, dividi g y (p ) , we h ve (D).

n 1 F mu a A a d B a n n m 0 . or l e ( ) ( ) f il, whe p 1 F mu a 0 a n m 0 . or l ( ) f ils, whe

F mu a a n 1 0 . or l (D) f ils, whe p

EX A M P L ES .

z 2 — 2 a V a x + -- s i n 2

2’ 2 x (a

A A maki n pply ( ) , g

= 2 n = 2 = d = a b m , , p ,

2 — 2 % 3 — i x ( a x ) a (aw Er e = — 2 — 2

( z 2 % ‘ l g(a x ) gsm g

A B mak n pply ( ) , i g

— 2 m = 0 n = 2 0 . a = = 1 a b . , , 1 é, ,

2 2 2 dx (a + x 2 2

2 2 2 % 2 a x l o x x g( + ) + % g ( + ) .

3 2 a 212 IN TEGRAL CA L C UL US.

Apply making

2 m — 3 n = 2 d = , , p a , b

2 * ' 1 " 3 2 2 l M ‘ 1 2 2 f x a x dx x a x dx. ( ) 2 2 ( ) 2 a 2 a

4 2 Ex . 05 . , p , gives

— dx 1 “ 1 2 2 I x (a 16 ) da: __ 2 2 2 w\/a _ x a — x

n b n m n Sub u a c . stit ti g, we o t i the o plete i tegral

5 2 “ 2 x ) ? 3 a ( a

A mak n pply (D) , i g

2 m = 0 n = 2 a = , , p a , b

Ex 180 . 33 . , p , gives

2 2 " ' a x gdx , 2 , ( a x )

Sub u n a stit ti g this, we h ve

dx 2 x

2 2 % 2 2 2 % “ 2 2 % (a x ) 3 a (a x ) 3 a (a x )

” 2 2 3 a ( a x ) 2 2 3 a 2 x

214 INTEGRAL CAL 0 UL US .

1 —1 x tan . 3 6 93 R fi x/2 V2

— 3 x .

2 2 x + 1 _2 a . 3 x3

A A and n a uc d to pply ( ) , the i tegr l is red e CHAPTER V .

TR IG ONO M ETR IC INTEG R A LS .

" ” Re ui re an x dx o r c o t x dx . 2 5 . q d ,

T m c an e n n n an n hese for s b readily i tegrated whe is i teger, positive o r negati ve .

“ ” ‘ 2 z tan x dx an x (sec x 1 ) dx

“ 2 z ‘ 2 x sec x dx tan” x dx

‘ l tann x n ‘ z tan x dx . n 1

Thu s tan" x dx is made to depend upon u ma b ucc uc n u n an x dx lti tely, y s essive red tio s, po

n n n a n a ak Whe is eg tive, the i tegr l t es the

” x o t x d ,

c c an be n a i n a m a mann whi h i tegr ted si il r er.

5 m n Fo r xa u ta x dx. e ple, req ired

5 2 tan x dx (sec 1 ) dx

3 an x dx .

s 2 tan x dx x (sec x 1) dx

l o g sec x.

an he 5 t Henc e an x dx lo g sec x . 2 216 INTEGRA L GAL OUL US.

Re ui re ec” x dx r ” 26 . o o s x q d , c ec x d .

T m c an be a n a n n hese for s re dily i tegr ted, whe is an even v n positi e i teger .

” ‘ 2 f see x dx ec” x sec x dx

n —2 2 z tan x 1 2 x ( ) s e c x d .

n n be a numb an d If is eve , will whole er, factor can be expan ded by the Binomial Theorem m n a c ter s i tegr ted dire tly. The w n x am follo i g e ple will illustrate the process .

ﬁ sec x dx

z 2 ‘ ’ (tan x see x dx (tan x 2 taufx 1) see x dx

5 3 ta x 2 tan x -- n x + l ta . g é

m " m ” 2 Re ui re an x x dx o r c o t x co sec x dx. 7 . q d see ,

These forms may b e readily integrated when n is a positive i m e n numb o r n m a o sit v e o dd nu b . ve er, whe is p er

r b n n n m o f A t . 26 a ca . Whe is eve , the ethod is ppli le This is illustrated by the following example

e 4 z z tan x sec x dx tan x 1) se c x dx

9 7 ta x tax s ﬁ ’ (tan x tan x)see x dx g ;

W n m o dd c as he is , pro eed follows

- m ‘ l n 1 an x see” x dx x see x sec xtan x dx

m—l H 2 ’ n x x 1 ) s co x sec x ta d .

218 INTEG AL R CAL C UL US.

W n n o dd he is ,

n ‘ 1 an x t n x ” t a tan x tan x dx + _ n — 1 n — 3 n — 5

7 5 t n x n s a 3 ta x 3 ec x dx t n x n a ta x . 7 5

5 3 s c o t 2 x co t 2 x o sec 2 x dx 10 3

7 5 4 ‘ tan x tan x au x sec x dx . 7 5

6 3 8 8 0 513 dx c o t x tan x —2 co t x 3

‘' 2 tangx 2 tan g g 4 x an x see x dx 5 9

6 s 5 ‘ c o t x c o t x o t x co sec x dx 6 8

’ 5 3 5 see x see x an x se c x dx 7 5

9 7 5 o se 5 5 c c x 2 c o sec x c osec x o t x co sec x dx . 9 7 5

2 s ’ 2 x 5 Z’ see an x sec f x dx 2 sec x 7

3 l 2 ’ au x co t x dx au x c o t x l o t n (t ) (t ) g a x.

7 5 3 tan x 2 tan x 2 tau x

4 3 3 19 . s x t n x dx x u x 4 s x x ( ec a ) g(see ta ) ec . TRIGONOMETRI0 INTEGRA LS.

i re c ” x Re u o s x d . 28 . q d

This is readily integrated when m o r n is a positive o dd m r n n n m nu b o m n a a n u b . er, whe is eg tive eve er d n Suppose n to be o d a d positive .

' " m 2 2 sin x o ” x dx in x in x o s x c s (1 s ) c dx.

n 1 As a o n c n ac r can be 5 is p sitive i teger, the se o d f to n m ex a and n a a a . p ded, the ter s i tegr ted sep r tely Fo r xam e ple,

2 5 2 f 2 sin x co s x dx in x (1 sin x) co s x dx

6 4 f sin x 2 sin x sin x) co s mdx

7 5 3 sin x 2 siu x sin x + 7 5 3

A m a c ma be u n m o dd and o ve . si il r pro ess y sed, whe is p siti Fo r xam e ple,

3 2 2 z sin x co s x dx o s x (1 co s x) sin x dx

f 4 (co s x co s x) sin x dx

3 5 c o s x co s x 3 5

n m n a n a n nu mb m an b e Whe + is eg tive eve er, the for c integrated by expressing it in terms o f see x and Thus

m s in x co s ” x dx dx

dx. 220 INTEGRAL CA L C UL US.

S nc m n and n m f i e is positive eve , the ethod o Art . 27 a cab is ppli le .

Fo r xam c n e ple, o sider

Here

EXA M P L ES.

5 7 ‘ 3 sin x u x in x co s x dx si 7

5 ’ g 5 4 c o s x 2 c o s x c o s x in x co s x dx 5 7 9

7 5 c o s x 8 x 3 0 3 c o s x c o s x. 7 2

5 5 10 3 0 8 —dx Si n Si n g 5 8 111 3

1 2 sin x . Sl n x

3 c o s x c o s x 4 5

5 co t x 5

4. 3 “a x 2 t n x ; a c o t x.

‘ 2 tan x 3 tan x 3 lo n g ta x. 2

2 5 Si n§ x dx 3 tan? x 5 COS

au x 3 co t x (t ) .

222 INTEGRAL GAL GUL US.

EX A M P L ES.

4 sin x dx sin 2 x

4 c o s x dx

z 2 s in x co s x dx

3 6 — sin 2 x 3 sin x dx x 4 sin 2 x + 3 4 e e Si n 2 x c o s x dx + Sin 4 16 3 §

4 4 sin 8 x s in x co s x dx x s1n 4 x

e o i 2 3 c s x s n x dx 5 x 2sin 2 x siu 4 x

s sin x dx

3 4 sin 2 x gsin 2 x + gsin 4 x

Int 3 0 . e ra tio n o T i g f r go no metric Functio ns by Tra nsfo rma ti o n i nto Al ebra ic Fun g cti o ns .

m ” in n a sin x co s x dx a um sin x = z If the i tegr l , we ss e , we have also

2 I ‘ l c o s a; 1 2 x sin z dx ( ) , ,

2 Henc e s in’” x co s” x dx

n —l

'" ? 2 z 1 z ( ) dz. TRIGONOMETRIC INTEGRAL S. 223

n o f mu a o f uc n m n By mea s the for l e red tio , this for is i te

all n a a u o f m an d n o r n a . grable for i tegr l v l es , positive eg tive In the prec edin g tran sform ation we might have assumed in c o s x z n a s x z. , i ste d of An x n c n a n n sin x an d co s x m a ca y e pressio o t i i g , free fro r di ls, c an u be n a b a mu a o f u c n o r b th s i tegr ted, either y for l red tio y n M n c n m un c a a za n . c r tio li tio oreover, si e the other trigo o etri f tion s c an be expressed ration ally in terms o f the sin e an d n a an a na n m c x n c osi e, it follows th t y r tio l trigo o etri e pressio n n c a be i tegrated .

EXA M P L ES .

3 5 2 4 co s x c o s x c o s S1n x co s x dx 12

2 2 A um c o s x z sin x 1 z dx ss e , ,

2 4 2 % sin x co s x dx (1 z ) dz.

B mu ae o f uc n y the for l red tio ,

8 z 4 2 % 1 z (1 (1 z ) 12 16

u u n z co s x w e a n al u . S bstit ti g , h ve the i tegr req ired

3 x n sec x dx W $10 g (sec ta x) .

dz ‘ l A um sec x z x sec z dx ss e , , z z\/z 1

2 3 ”‘ s ec x dx z— w/z 1 10 2 1 [ Vig I g 3 53 0 )

s ec x tan x lo x t n g (see a x) . z é

dx 1 l o tan 2 g sin x co s x co s x 224 I NTEG AL C CU S R AL L U .

dx sin x 1 3

10 sec x tan x . 2 3 z g ( ) siu x c o s x 2 c o s x sin x 5

c o s x 3 _. co s x - l0 an 2 g t 2 8 1n x 2

0 2 3 Sl n x dx sm x sm x

5 4 z co s x 4 c o s x 8 co s x 8 A um tan x ss e z. dx 1 mmx — l 4 w x tan a lo co t x n g ( ta a ) . tan x

dx bx a 10 a sin x + b co s 2 2 z 2 g ( a tan x + b a + b a + b

Tri o no et Fo rmulae o e u i n m ric R ct o . 3 1 . g f d

’” B m an o f n mu a sin x co s" x dx ma y e s the followi g for l e, y be

b a n all n a a u o f m and n b suc c r o t i ed for i tegr l v l es , y essive e n ductio .

m ” sin x c o s x dx

m “ l n 1 sin x co s + x m 1 mm“ ? " 8 x co s x dx. m n m n

1 ” c o s” + x m —n 2 c o s x dx

m ‘ l — m‘ 2 (m 1 ) sin x m l s in x

sin’" x co s" x dx

m+ l n —l 8 m x 0 S x o '” ’ “ 2 sm x co s x dx. m n

x n m 2 sin’” x dx

‘ 1 " “ 2 (n 1 ) c o s ” x c o s x

—1 sin” s in’” x dx

226 I C CU US NTEGRAL AL L .

EXA M P L ES.

5 6 c o s x in x 5 i —- 3 § n x dx siu x sin 2 3 12 8

5 COSx o sec x dx

7 sin x 1 5 s x 3 . ec dx 2 4 z 2 co s x 3 c o s x 12 c o s x

5— log (see x tan x) i g

8 7 o s x clx o s x + g c o s x

5 c o s x in 3 4 ’ x siu x in x co s x clx 2 3 12 8

‘ 2 z o s x s in x co s x gsi

COS x z (3 co s x 2 sin x 2 1 1 5 g z 3 c o s x 3 sin x 3 sin x 2

glo g (sec x tan x)

dx 33 R agui red a b S111 x

m 2 § 2 § a + b sin x = a o s + sin + 2 b sin co s 2 2 2

z i’ sec gdx a sec gdx

a + 2 b tan g+ a tan

where z a tan g b. L TRIG ONOMETRIC INTEGRA S.

If a b num ca , eri lly,

0 . b sin x

a b num ca If , eri lly,

d + b sin x

Re ui red 3 4 . q at b c o s x

= l’ - 2 d + b co s x a o s gF sin + b os — b)sin

2 sec gdx — d + b + (a b) tan

ut tan z If we p ,

dx dz

- — 2 a + b co s x d - b ( a b) z

b num c a If a , eri lly

dx 2

a b c o s x I CAL C UL US NTEGRAL .

a b num ca , eri lly,

dx 2

a + b co s x b —a

2 2 6 “ “ / — — x b a tan g Vb + a

“ “ 3 5 . Re uire e sin nx dx and e co s nx q d , dx.

“ n a n b a u e I tegr ti g y p rts, with , “ “ e c o s xx a “ e s1n nx dx e co s nxdx. n a

n a n am u sin nx I tegr ti g the s e, with ,

‘” c “ c sin nx n “ m nx dx e co s nx dx e s . a

E m na n m 1 and 2 eu co s nx dx a e li i ti g fro ( ) ( ) , we h v

2 ? “ ( d n ) e sin nx dx sin nx n co s

m e a n nx n co s nx az si ) e sin nx dx f “2

Sub u n in 1 and an n s stit ti g this ( ) tr sposi g, give

2 “ sin nx a c os nx) e co s nx dx 2 2 (d + n ) n

“ e n s1n nx a c nx u ( os ) e co s nx dx

A CH PTER VI.

NTEGR A L-S F O R R EFER EN E I C .

f r n 3 6 . We give o refere ce a list o f some of the integral s n th e precedi g c hapters .

1 .

A EX PO NENT I A L INTEGR L S.

a t:

lo g a.

6 .

E RA S TR I GONOMETR I C INT G L .

in x dx c o s x .

0 8 x dx sin x.

an x dx l o g sec x.

n x dx l o g si x. INTEGRALS F OR REFERENCE. 231

s ec x dx lo g (sec x tan x)

log t

osec x dx lo g ( c osec x co t x)

l o g tan g

z x dx t n ec a x.

2 o sec x dx co t x.

e c x tan x dx sec x.

o sec x co t x dx x c osec .

? x 1 in x dx sin 2 x. 2 4

2 x 1 18 . co s x dx l n 2 S x. 4

2 TE RA S C INING —x IN G L ONTA . 232 INTEGRA L CAL C UL S U .

2 z z 2 2 V a x dx = \/d x + 9—si n g 2

— z = 2 — 2 z d x dx g(2 x a ) v a

2 2 2 2 ( d a \/a £17

z 2 % 2 2 3 a (a x ) dx (b a x + — i sin g 8

INTEGR A LS C ONTAINING

IN T GRA L CA L C L US 234 E U .

2 3 3 0 . + 6 2 ”ﬁ — 2 / — 2 d x x dx ? x 2 d x x + %vers

2 INTEGR A LS C ONTAININ G j : d x bx

“ 2 a x b tan

. 1

2 2 — \/b —4 a c 2 ax + b + V b 4 ac

60 . INTEGRAL S F OR REFERENCE. 235

TH E NT EGR A O R I LS.

dx = n V ( d (a. b) si A PTER CH VII.

M M IN TEG R A TIO N A S A . S U A TIO N . DEFINITE

NTEG R A L I S .

The c o f n ra n ma b e a as 3 7 . pro ess i teg tio y reg rded the

u mma o n o f an n n o f n n m m s ti i ﬁ ite series i ﬁ itely s all ter s . u o n c n n b m A n a . s a ill str ti , o sider the followi g pro le

To ind the a rea 38 . f PAB Q i ncluded between a iven curve OS the g ,

a xis o a nd the o r i f X, d

n a tes AP a nd B Q. % Let y x be the equ a

tion o f the given curve .

Le t A a B b. 0 , Q Suppose AB divided A AI A2 in to n equ al parts (i n the u n an d Ax ﬁg re, let

f ua a as AA A A den ote o ne o the eq l p rts , ] , 1 2,

x Then A B b a nA .

At A a n a A P A P m an d c o m A 1, 2, m , dr w the ordi tes l l , 2 2, ,

lete c an PA P A p the re t gles I, I Q, % o n c u x From the equ ati Of the rve, y , % a % P A = a Ax P A a 2 B b PA , 1 1 ( )? 2 2 ( Q A AA a Ax. Area o f rectangle PAl PA x 1

P A A A a Ax Ax. Area o f rectangle P 1112 I 1 x I 2 ( ﬁ = P A A A a 2 Ax Ax. Ar e a o f rectangle P2A3 2 2 x 2 3 ( ﬁ

238 INTEGRAL CAL C UL US.

T a ” dx % h t is, 20 )

‘ 2 nc xi dx % % He e, x dx x § ( ) 5

Sub u n x a a dx a 2 dx m b dx stit ti g for , , , , , ,

g' % we have a dx § (a dx) ga i

% (a de ﬂate g(a 2 dx) g. (a

2 b e ant b% — b dx % ( d ﬁ 5 § ( )

A di n an d canc n m in n m mb w e d g elli g ter s seco d e er, have i — % % a dx + dx) dx § b

T a as x a m a b sum o f ucc h t is , v ries fro to , the the s essive

nc m n o f unc n xg u a n nc m n i re e ts the f tio g is eq l to its e tire i re e t .

b % % g” x dx § b ga area PAB Q .

W e have thu s shown that the sum of the inﬁnite series repre

n un u u n x b and a i n se ted by is fo d by s bstit ti g for , i ﬁ the inte ral o f x dx an d s ubtracting the latter result from g , g , m the for er . b 1 " ' 2 e nite i nte ral and The expression x dx is called a d ﬁ g , the

tin between limits proc ess o f evalu ating it is c alled i ntegra g , the n i n eri o r limit and i nitial valu e a o f the variable bei g the f , the

final value b the sup erio r limit. f In contradi stinction fi is called the indefinite i ntegral O t g x dx. INTEGRA TION A s A S UMMA TION . 239

b i The a n o f m x dx 40 . rel tio the ter s of the series to the

73? integral 2x may be made clearer to the studen t by c onsider ing the following series Of numbers 1 3 4 5 9 7 16 9 25 11 36

The numbers in the second column are the differenc es be n c n cu numb i n an d n a twee o se tive ers the first, it is evide t th t the sum o f the secon d c olumn o f numbers is the differen ce b n nd a i n umn T a c . a etwee the first l st, the first ol h t is, —1 .

b ‘ The m o f xi dx ma be m a a an as f ter s y si il rly rr ged, ollows

dx %dx (b ) . 240 NTE RA L L I G CA C UL US .

%" g’ S nc x dx ff n a o f x m in c n i e is the di ere ti l g , the ter s the se o d c olumn are the inﬁn itesimal differenc es between the c o n se cu

m in t an d r tive ter s the ﬁrs , the efore

) i 1 a xAdx b:T 2 3 th t is, 06 .

Genera l ni tio n o a e n te Inte r l a . 4 1 . Deﬁ f D ﬁ i g

In n a < ) x n an n unc n o f x hi c i ge er l, if 1 ( ) de ote y give f tio , w h s

n an d c n nu o u m x a x b x dx fi ite o ti s fro to , ¢ ( ) is the definite integral represe nting the su m of an infinite series o f m b a n m b u n x a m ter s, O t i ed fro y s pposi g to v ry fro a to b.

< ) x dx x n n n a If 1 ( ) 30 ( ) , the i deﬁ ite i tegr l,

— tlf (a)

r 8 b u T ma b e u a b an a a as in A t . 3 his y ill str ted y re , y s ppos in x be ua o n o f cu OS and ro o g y ¢ ( ) to the eq ti the rve , the p f

m m m b ub u n > x o f A rt . 3 9 a b e a y si il rly odiﬁed, y s stit ti g d ( ) for

g’ and z/1 (x) for gx .

add in a c o f a o n b n 4 2 . We this rti le the proof the rel ti etwee n nd n n n a x i n m o f the deﬁ ite a i deﬁ ite i tegr ls, e pressed the for

l in A rt 39 . limits instead of in ﬁnite sima s as . mit ” n We shall u se the expression Li Am O to de ote the words “ ” Th e m a Ax a ac z o o f . li it, s ppro hes er , d Gw n w w e e ( ) i ( ), a;

2 2d) (x) Ax =

> b —A x Ax <1 ( ) ,

242 INTEGRAL CAL C UL Us .

nc s Ax an c a Ax. He e 2 v ishes with k, th t is, with

Taking the limit o f we have

b “ Limit x Ax x dx M ) W ) Ax = 0 2 2¢ ( ) cf) ( ) .

b e n c a a b a c n an c in the 43 . It is to oti ed th t the r itr ry o st t , m n n n n a a a n a . i deﬁ ite i tegr l, dis ppe rs fro the deﬁ ite i teg r l

3 u in a ua n x dx ca n n n r Th s, if ev l ti g , we ll the i deﬁ ite i teg al

c a , we h ve

4 4 4 5 b 91 + 0 as b r . 2 , efo e

Or if in eval uating we call the indeﬁnite inte

ra x 0 a g l d( ) + , we h ve

b 0 ? a 2( ) + 1N ) M ) .

f as be ore .

EXA M P L ES.

Evaluate the followi ng deﬁnite integrals

3 4 64 1 _x _ 3 1 3 3

6 = 1 lo g x log e lo g 1 . l

sin x dx — co s x = O

2 3 b x dx 4 . ( ) DE I ITE I TEGRAL 243 F N N S.

3 x dx lo g 2 . 1 + x2 2

3 8 a dx

2 2 x 4 a

4 sec 0 d9

dx 7r —a . 2 x 2 x COS a + 1 2 sin a

dx 1r . 2 2 2 (d + x ) (b 2 a b(a b)

’“ ef sin nx dx i n 2 n

co s x 3

n b n A rt Derive the followi g y (5) a d . 31

15 . n i s n If eve , — 1 -3 o 5 u - n 1 r ' ( ) $ 1113 a} dm COS wdw N . n 2

16 . n If is Odd,

— 2 o 4 ~ 6 m (n 1 ) 8 111 x dx OS x dx i 244 I NTEGRAL CAL C UL Us .

h u in . n ab 4 3 5 C a nge of L imits . Whe a n ew vari le is sed

b a n n n n n a w e ma a o a n o t i i g the i deﬁ ite i tegr l, y v id the restor tio o f o na a ab b c an n m c n the rigi l v ri le, y h gi g the li its to orrespo d

with the ne w variable .

Fo r xam a ua e ple, to ev l te

4 dx assume 0

dx 2 z dz Then we have 1 z 1 + v Q +

N h n x = 4 = n h n x = = O o w w e z 2 a d w e 0 z . , ; ,

4 2 2 dx 2 z dz — Hen c e Ez lo s (1 + z 0 1 + ﬁ 1 + z — — 4 2 lo g 3 .

EXAMPLES .

Assume x

3 A u m x — 2 = z ss e . 0 (x

lo g s x x _ e e 1 z ” — = z dx = 4 A ssu me e i . ex + 3

dx 1o g (se0 a + t an (1 ) Qx —- z an a X/e i tan a t h 2 2 a A ssume e tan z .

7 f ( sin d c o s 9) d0 lo g 3 . m in c o s 6 x Assu e s 0 . 3 + s in 2 0 4

2 ( x 1 d“; A um x z. l o 3 . ﬂ . g ss e “ 2 x\/x - 7 x 1

246 NTEGRAL CAL C L I U US.

ma a a u a a as n a b We y lso reg rd the req ired re ge er ted, y the na A P m n m an d a n i n n ordi te ovi g fro left to right, v ryi g le gth acc n t o ua n n u R a n as ordi g the eq tio of the give c rve . eg rdi g y

c n an m n anc dx n a re c o st t while ovi g the dist e , it ge er tes the T n n a o mu u tangle ydx. he the ge er l f r la for the req ired area is

A dx as b y , efore ;

n m a A n n n a n o f the i ferior li it 0 , de oti g the i iti l positio the m n na an d u m b OB n a ovi g ordi te, the s perior li it , its ﬁ l posi n tio .

S m a a n n cu ax i il rly the rea betwee the give rve, the is Of Y, and n ab c a two give s iss s , is

the limits o f integration being the limiting values o f y.

EXA M P L ES.

2 ‘ n n a ab a 4 ax and axi s 1 . Fi d the area betwee the p r ol y the

m n na at n h It . Of X, fro the origi to the ordi te the poi t ( , )

3 h % g‘ % % 4 a h y dx 2 a x dx 3 0 3

2 t S nc 76 4 a h k 2 a n . i e ,

- n A hk two c cum c b c a g . g , thirds the ir s ri ed re t le 2 10 - : b. ns . tra F n n a a 4 1 . A i d the e tire re of the ellipse 2 3 a 2

Show that the area o f a sector o f the equ ilateral hyper 2 2 2 bo a x a nc u b n ax and a l g , i l ded etwee the is Of X am u n x o f cu di eter thro gh the poi t ( , g ) the rve, is 2 a w + y lo g o 2 a 3 8 a F n n a a b n c , i d the e tire re etwee the wit h y 2 2 93 4 a

and the axis o f X. ﬁ 4 . A ns . wa A REA S OF C URVES 24 . 7

n a a n c - n 5 . Fi d the re i ter epted between the co ordi ate axes by

’ 2 ab a i i ? the par ol x y a a“ A ns . 6

F n n a 6 . i d the e tire rea within the curve

Aﬂns n a b.

§ il F n n a a n 3 a i d the e tire re withi the hypocycloid x y . 2 3 7ICL

A ns . 8

3 2 96 Find the entire area between the cissoid y 2 a x and n x 2 a a m the li e , its sy ptote . g A ns . Swa .

The a a b n cu sum o r f nc re etwee two rves is the , the di fere e, o f the areas between the cu rves and o ne o f the c o -ordi na ax m b n m n b n te es , the li its ei g deter i ed y the poi ts Of n n i tersectio .

2 F n a a n u b n ab a x : 4 a i d the re i cl ded etwee the par ol g, 3 8 a I 2 an c a . d the wit h y W A ns . 2 Z r x + 4 a

Areas o urves . 46 . f C

o lar o - r in t s To n P C o d a e . ﬁ d the a rea P O i nclu e be Q, d d

tween a iven curve P a nd g Q,

two iven r a ii vecto res P g d , O a nd Let OQ.

P OX OX 8 . Q ,

Let r and 0 b e the c o -o r n n di nate s Of a y poi t P 2 of cu n the rve, the

r Ar AO + , 6 + , will be the c o -ordinates o f P3 . 248 INTEGRA L A C L C UL US.

Th e a a o f c cu re the ir lar sector P 2 OR 2 is 1 1 —. ' 1 0 P 2 X P 2 R2 7 ° TA 9 § 2 2

Th e su m o f c o P OR P OR P OR ma be the se t rs , I l , 2 2, m , y rep n b rese ted y B ﬂna

a. é

Th e u a a P O m o f sum c req ired re Q is the li it the Of the se tors,

as AO a ac z . T a ppro hes ero h t is, —1 B A:

4 7 ma a a a a P O as n a b . We y lso reg rd the re Q ge er ted, y the a u c o n m OP to O an d a n i n n r di s ve t r revolvi g fro Q, v ryi g le gth o n to ua n o f n u ac c rdi g the eq tio the give c rve . R a n r as c n an c b n an dd eg rdi g o st t while des ri i g the gle , it gen erates the sector whose are a is éﬁ dd. B 2 n c A 1 r d9 as b He e , efore ;

n m no n n a and u mi the i ferior li it a. de ti g the i iti l, the s perior li t m n u na n a c . 6, the ﬁ l positio , of the ovi g r di s ve tor

EX A M P L ES .

Find the area des cribed by the radiu s vector in o ne entire a f A m r a revolu tion o f the spir l o rchi edes 0.

2 2" 2 3 2" 3 2 1 ’f 1 2 2 a 0 4 rt a Here A _ r dd a 0 d0 2 3 0 3

Find the area des cribed by the radiu s vector i n the loga “ 17 mi a r e m 0 O 0 . rith c Spir l , fro to

m (e

250 INTEGRAL CA L C L U US.

EX A M P L ES .

2 1 . F n n o f are o f a ab a 4 a x i d the le gth the the p r ol y , from x x m o f a u u the verte to the e tre ity the l t s rect m.

dx x3

therefore s 1 dx

In a b 9 21 m n . 3 ak b O tegr ted y , p , i g .

i 2 dx = x/a x + x + a lo g

Fin d the len gth o f the arc of the semi-cubical parabola 2 3 d x m n x 5 a . 35 v , fro the origi to 3 a A ns . 2 7

2 F n n o f arc o f cu 9 a x x 3 a i d the le gth the the rve y ( )f, m x = x = n 2 a / O 3 a . A s . 3 fro to \ .

a “ 4 F n n o f arc ca na e e . i d the le gth the of the te ry y gv

m x 0 n x . fro to the poi t ( , y)

A ns .

Fn n n arc o f c c o 5 . i d the e tire le gth Of the the hypo y l id

A ns . 6 a .

- 4 9 L en ths o Curves . P o lar Co o r ina tes . To nd the . g f d ﬁ t een two iven o ints P a nd length of the a rc P Q be w g p Q .

: Let P X OX e . O Q , LENGTH s OF CUR VES. 251

g’ dd

A r 3 t . 98 . al . ( ) 2, Dif C therefore

2 7 + the limits being the limiting valu es

Of 0.

Or we have ds

f dr therefore ,

m m n = b n a u o f r . T a OP a the li its ei g the li iti g v l es h t is, , OQ = h

EX A M P L ES.

Fin d the length o f the arc o f the spiral o f Arc himedes

m o o nd o f n r a t e u . d, fro the p le the the ﬁrst revol tio

d r dd

r 2 2 % i (a e a ) dd d (1 as de

2 élo g w 9 )

2 d + 4 7 + %l o g (2 7r

F n n n 1 i d the e tire le gth Of the c ardioid r a ( c o s d) .

A ns . 8 a .

" F n n o f o a m c r a r ea o m i d the le gth the l g rith i spi l , fr the o n m r d . U se u a pole to the p i t ( , ) for l

I " A ns . v a 1 . a 252 G CAL C UL s INTE RAL U .

3 4 F n n n o f cu r a n . i d the e tire le gth the rve si g

3 ” a

A ns . 2

Th u n o f i 5. e a o c c a u x eq ti the epi y lo d, the r di s of the ﬁ ed

c c b n a an d a o f n c c 9 ir le ei g , th t the rolli g ir le , is 2

*‘ 2 27 a r

From the above equ ation

2 2 d0 2 \/r _ a n m n u a . the se For ula A s . 6 ‘ 2 d 7 TX/4 a

h v lum ur a es o e luti n . To nd t e o e S c R vo o o lume . 5 0 . f f V ﬁ

enerate b revo lvi n a bo ut OX the la ne a rea A P B . g d, y g p Q

Le t a B b. 0 1 4 , Q Let a; an d y b e the — co ordinates o f any point n u P 2 of the give c rve . It is evide nt that the

rectangle P gAzrél3 w ill generate a right cylin u m der, whose vol e is zA wy w. Th e su m o f all these cylinders may b e repre A b A3 4 8 z sented by w z y Aw.

sum c n The requ ired volume is the limit of the of the yli ders,

z T a as Ax approac hes ero . h t is ,

V = 7r

Or we may reg ard the required volume as generated by the

a a o f a c c c m an a a e r endi cu~ re ir le, whi h oves with its pl e lw ys p p

' 1ar to ax o f c n m n a n ax and the is X, its e tre ovi g lo g this is, its radius being the ordinate o f the given curve .

254 IN TEGRA L CAL C UL S U .

E A M L X P ES.

Fin d the volu me and su rfac e o f the prolate sphero id 2 2 3 o b a n b n ab u X l 3 g t i ed, y revolvi g o t the e l ipse 1 . 2 2 a b

F m 1 A rt 0 . 5 ro ( ) , we have 1 “ a ) doc 2

From

b2m2

2 x )

d 4 2 ? 2 l [a (a b ) x ] dw

8 111

F n th e um and u ac n a b e n u i d vol e s rf e g e er ted, y r volvi g abo t 2

. the a ab 4 a x r m o n to X p r ola y , f o the rigi a: a . — g 8 5 1 ) A ns . 2 7ra and 33

Fin d the volume an d c onvex surfac e o f the right con e

n a e b o i n ab u X n o n n ge er t d, y rev lv g o t the li e j i i g the 2 i i n and o mt a b . wa b or g the p , 2 2 ( ) n / A s . an d 7 rb\ a + b

F n n o um and u ac n a b n i d the e tire v l e s rf e ge er ted, y revolvi g ii gi i about X the hypocycloid x y a

A ns . 2 OTH ER VOL UMES. 55

F n n um n a b n c 5 . i d the e tire vol e ge er ted y revolvi g the wit h

2 3 Ans 4 a . . 1r 2 2 x 4 “

F n u me n a b n ab u a i d the vol ge er ted y revolvi g o t X, the p rt

‘ l l’ 5 o f a ab a w a n c b c o - i the p r ol + y , i ter epted y the ord

?’ n x n a ate a es . A ns ° 15

Find the volume and s urface o f the torus generated by ? 2 2 n b u a a c c b a . revolvi g o t X, the ir le (y ) 2 2 2 ns 2 n d 4 a b A . 7r a a 1: b .

Find the volume and surface generated by revolving

= z ab u ca na e + e m a 0 m a . o t Y, the te ry y g( fro to

3 wa _ — 1 — 2 - 1 (e + 5 e 4) and 2 7ra (1 e 2

the Th m f n n um o f O r o lumes . e o 5 2 . V ethod ﬁ di g the vol e a o f u n in n n n t A rt . 50 b c a solid revol tio , y o sideri g it ge er ed . b a m i n c c o f a n a u ma b e x n an y ov g ir le v ryi g r di s, y e te ded to y

' a a o f a n c an be x as a solid, Where the re sectio e pressed unc n o f n ance m a x n f tio its perpe dicular di st fro ﬁ ed poi t .

n anc b a and If we de ote this dist e y , the area o f the section b a um y X, we h ve for the vol e, 256 INTEG A L CAL R C UL US .

E M P L X A ES .

Find the volume o f a pyramid o r c one having any base a wh tever .

Let A be a a o f ba and h a u the re the se, the ltit de . Let a; n n cu a anc m x de ote the perpe di l r dist e fro the verte , o f c n a b a a n a se tio p rallel to the se . C lli g the area o f c n as i n a b m this se tio X, we h ve y solid geo etry,

2 2 X x A n; ix A hr 7 ?

Substituting in A h3 Ah h2 3 3

um o f a c n c cu a ba vol e right o oid with ir l r se, the h o f b b n a and a u . as e ei g , ltit de

0 A = B 0 = 2 a B 0 = 0 A = h

‘ The c o n R T n cu a se ti Q, perpe di l r to

A an c an . G , is isos eles tri gle

Let a: OR; then

2 = P T =h 2 a w—a X area R TQ X P Q V .

Substituting in we have

This is o n e-half the cylinder o f the s ame base and a u ltit de .

A an m m a x n o ne a n rect gle oves fro ﬁ ed poi t, side v ryi g as n m n and h as the dista ce fro this poi t, the ot er the u f n At anc o f 2 sq are o this dista ce . the dist e feet, a the rectangle bec omes a square o f 3 feet . Wh t is

47 cub c . the volume then generated Ans . } i feet

C A TE H P R IX .

E N E SU C C S SIV E I T G RA TIO N .

Do uble Inte ra l. r a n r r 53 . g If we reve se the Oper tio s ep e 8 2 n b a a c a a o uble i nte r se ted y we h ve wh t is lled d g al. 693 gy

i 3 Fo r xam u w e ple, s ppose g/ ,

2 3 n 1 d dx the 3 ?! ,

c n ca u c c n a n ﬁ whi h i di tes two s essive i tegr tio s, the rst with nc a a n as a c n an and n refere e to , reg rdi g y o st t, the seco d

nc a n a: as a c n an . T u with refere e to y, reg rdi g o st t h s

_wiyf ’ 12

omitting the con stants o f integration .

e nite o uble Inte ral. n a n are 5 4 . D ﬁ D g Here the i tegr tio s m between given li its .

Fo r x am e ple,

a 2 (a at) y dy da: d y

2b 2 2 3 7 a b a y2dy 2 6 “ In ab a x dx n a n the ove ( )d y , the right i tegr l sig

m 0 and a b e u a ab x and with the li its , is to sed with the v ri le , m b an d 2 b a ab a the left with the li its , with the v ri le y ; th t is, the integral signs with their limits are to b e tak en i n the s ame f n a d dx at end and m ri ht order as the dif ere ti ls y, , the , fro g to SUCCESSIVE IN TEGRA TION. 259

S m m m o f n a n are unc 55 . o eti es the li its the ﬁrst i tegr tio f n n tio s o f the variable of the sec o d . Fo r xam e ple,

3 2 — 2 <3 y + 2 ay a y) dy

A s an xam other e ple,

(90 + y) dx dy

Tri l n A e I te ra l . m n n u 5 6 . p g s si ilar otatio is sed for three ucc n n T u s essive i tegratio s . h s

a 2 2 x y z da:dy dz

3 3 (a O) .

EXA M P L ES.

Evaluate the following deﬁnite integrals

2 2 av a; a 9 g/( y) d dy g( a b) .

a s 3 2 a b r sin 0 dr d9 (co s B co s a )

262 INTEGRAL CAL C UL US.

Henc e

' ' 2 ? Moment o f MNN M dw (a: y ) dv

2 bx dx.

a n u un m m n o f a c a r ma H vi g th s fo d the o e t verti l st ip, we y sum all b u n a: in u these strips, y s pposi g this res lt to vary from O a . T a to h t is, 3 ' 2 } a b Moment o f OA OB x da: g 23

But the prec eding operation s are the same as those repre n b ubl n a se ted y the do e i tegr l,

2 ? x dcc d . ( y ) y (See Art .

c c all m n i n a ho rizo nta l and If we ﬁrst olle t the ele e ts strip , n sum z n a a the these hori o t l strips, we h ve

2 ? Moment o f OA CB (m y ) dy da: 0

To ﬁnd m m n 59 . the o e t o f in ertia of the right tri n a gle OA C about 0 .

L : et Ct A . 0 A , O 6 The equation o f 0 0 is b

This differs from the pre ceding problem only in the limits n n In n m n i n a c a o f the ﬁrst i tegratio . c ollec ti g the ele e ts verti l m O M B ut M n n MN a o N. N o a. strip , 3; v ries fr to is lo ger

n n n r 58 b ut ar M acc n c a as i A t . O o st t , v ies with , ordi g to the

u f ay H ence m o f are 0 and eq ation o (9 g . the li its y

In n l a b n n a n c ollecti g a l the vertic l strips y the sec o d i tegr tio ,

a: a m O a as in A rt . 58 . v ries fro to , AP P LICA TION OF DO UB LE INTEGRA TION.

a ’ 2 ? Moment o f OA O (or g ) dwdy a b

By su ppo sing the triangle c omposed of ho ri zo nta l

' HK a ﬁnd , we sh ll

Moment of OA G

2 2 (x y ) dydx

. Plane A rea a s a o u l in A r m 60 b e Inte ra l. 5 D g If t . 8 we o it ac a a n a o f m m n the f tor we sh ll h ve i ste d the o e t, the a a n u a re , of the give s rf c e .

T a A a dwd h t is, re y

m n m n the li its bei g deter i ed as before .

EX A M P L ES .

F n m m n f n u 1 . o a ab n o f i d the o e t i erti o t the origi , the right tr1an gle fo rmed by the c o -ordinate ax es and the line n n O o n a O b . j i i g the poi ts ( , ) , ( , ) — b ( a z )

A ns . 12

' n m n f n u n o f c c 2 . F m o a ab i d the o e t i erti o t the origi , the ir le 2 _ 2 4 " 0“ 2 2 7 4 x dx d 1 . A ns . ( + y ) y g

Find also the area o f the preceding c ircle by A rt . 60 .

A ns .

n b n a a n and a F b Art . 60 a a i d y , the re etwee str ight li e

a ab a ac o f c n n and o n p r ol , e h whi h joi s the origi the p i t

a b ax b n ax o f a ab a . ( , ) , the is of X ei g the is the p r ol 264 I TEGRAL CA L L N C U Us .

F n m m n o f n a f 5 . i d the o e t i erti o the preceding area about 1 1n the o r g . 5 £ . A ns .

F n m m n o f n a u n 6 . ab o f i d the o e t i erti o t the origi , the area 2 nc u n a ab a = 4 a m n w = a i l ded withi the p r ol y , the li e + y 3 , n x f a d the a is o X.

3(F a 2 ? 2 n x da' d ? A s . ( y ) y ke/3 y ) (la: dg

4 2 2 3 14 a x + y )dy dx 3 5

d l - o u ble Inte ratio n. P o la r Co o r inates . To ﬁnd 7 61 . D g d the B a a o f u a an o f a c c A O a u a . re the q dr t ir le , whose r di s is In c an u a c o - d n a re t g l r or i tes, n f A rt . 58 o n , the li es divisio c n o f m fo r o ne o sist two syste s, o f c a: c n an and whi h is o st t, for

c n an . the other, y is o st t So i n a c o - na pol r ordi tes, we have o n e system o f straight lines u fo r ac o f thro gh the pole, e h c d c n an and an whi h is o st t, other s m o f c c ab u as c n ac o f c r yste ir les o t the pole e tre, for e h whi h n is c o stant . Let r d c are be a as n n n a ab , , whi h to reg rded i depe de t v ri les, b e c o - na o f an n o f n c n as P and the ordi tes y poi t i terse tio , - n T n f P r dr d dd c o a o f . a a o , , the ordi tes Q he the re Q is

P R x R Q r dd dr .

n a a n d c n an r a If we ﬁrst i tegr te reg rdi g o st t, while v ries m O t o a c c all m n i n an c M OM fro , we olle t the ele e ts y se tor

Th e c n n a n um all c b a n d se o d i tegr tio s s the se tors, y v ryi g

fro m O t0 2

Hence Area B OA

266 I TEGRAL CAL C L N U US.

EX A M P L ES.

Find the moment o f inertia about 0 o f the area o f the 4 m A r c c i n t . 63 a se i ir le . 3 7r Ans 4

F n m m n o f n a ab u o f a a i d the o e t i erti o t the pole, the re

2 nc u b a ab a r a n a n OK i l ded y the p r ol see g, the i iti l li e , and a n at an u li e right gles to it thro gh the pole .

2 9 u ses z s 48 a r dddr 35 ?

F n m m n n a ab u c n r o f a a i d the o e t of i erti o t its e t e, the re 2 2 “ n a of o ne loop o f the lem is cate r a 0 0 s 2 d. w Ans 16

Fin d by double integration the entire area o f the c ardio id

r a i co s d . ( ) Ans 2

n m m n o f n a ab u the o f a a o f Fi d the o e t i erti o t pole, the re 5 the precedi ng cardioid . 3 A ns 16 CHAPTER XI.

S U R F A C E A ND V O LU M E O F A NY S O LID .

TO in the a rea O a n s u r a ce who se e ua ti o n i s iven 6 4 . f d f y f , q g

- etw en three recta n ula r co o r ina tes w z. b e g d , , y,

Let this equation be x f ( , y)

Suppose the given s urfac e to b e divided into elements by two n X Z an d Z a a a c Y . series of pl es, p r llel respe tively to These planes will also divide the plan e X Y into elementary

c an o ne c P c n u n re t gles, of whi h is Q, the proje tio po the plane ' ' Y o f c o n n m n o f u a X the rrespo di g ele e t the s rf c e P Q . ' Le t x z b e co - di na o f P and m dv d , y, , the or tes , + , y + y, ' z dz , of Q . 268 INTEGRAL OAL OUL Us .

w ’ ’ S nc P c n P a a o f P i e Q is the proje tio of Q , the re Q is equal ' ’ to a P mu b c n th t of Q , ltiplied y the osi e o f the inclination o f

’ ' ‘ P an T an Q to the pl e X Y. his gle is evidently that made by ' an n an at P an n in the t ge t pl e with the pl e XY. De ot g this an b gle y 7 )

' ' A a P Ar a P c o s re Q e Q y,

' ' A a P A a P e ‘ re Q re Q s c y.

We see from the ﬁgure that

A a P an re Q d dy.

A m ana c a m o f m n n lso fro lyti l geo etry three di e sio s, i

sec See . 7 , ( p

z 3 Q and Q are a a f n a c c n ak where p rti l di fere ti l oefﬁ ie ts, t en from an dy f n u a n o u ac z x . the eq tio the give s rf e f ( , y)

' ' Henc e Area P Q

n u u ac If S de ote the req ired s rf e,

m f n a n n n u n c n o n the li its o the i tegr tio depe di g po the proje tio ,

an o f u ac u . the pl e XY, the s rf e req ired

Fo r xam u u ac AB C be o ne- 65 . e ple, s ppose the s rf e to eighth o f the surfac e o f a sphere whose equation is

‘z 2 - 2 z a + y z a .

dz a: dz

270 INTEGRAL GA L OUL Us .

Find the area o f that part of the surfac e

2 2 2 z a c a s in a a ( os y ) ,

which is situated in the positive c ompartment o f co n ordi ates .

The u ac a c cu a c n ax s rf e is right ir l r yli der, whose is is the

n z = O mCOS a sin a = O li e , + y ,

n f a d radius o base a .

A am o f a S a i u a ax o f a di eter phere, whose r d s is , is the is m a u a ba 2 b b n o f right pris with sq re se, ei g the side the F n u ac f n square . i d the s rf e o the sphere i tercepted by m the pris . b ‘ 1 1 ns 8 a b n a n A . si si

TO nd the vo lume O a n so li bo un e b a sur a ce 6 6 . ﬁ f y d d d y f , who se e uati o n i s iven between three rectan ula r co -o r i na tes q g g d ,

2 .

A s a an a a b di n n m n a c an pl e re , y divi g it i to ele e t ry re t gles,

an ma be u be b an a a so y solid y s pposed to divided, y pl es p r llel c o - na an n m n a c an u a a a to the ordi te pl es, i to ele e t ry re t g l r p r llelo i eds The u m o n e a a doc d dz p p . vol e of of these p r llelopipeds is y , and the volume o f the entire solid is

a:d dz y , the limits o f the integration depending upon the equation n n u o f the bou di g s rfac e . OL UME OF ANY SOLID 2 V . 71

Fo r x m - 6 7 . a us ﬁn m f n e ple, let d the volu e o o e eighth o f e u the llipsoid, whose eq ation is

2 2 2 x y z 2 2 2 a b 0

P Q represents o n e o f the elementary parallelopipeds whose volume is dcc dy dz. n nc z c c all If we i tegrate with refere e to , we olle t the elements in the c olumn MN z varying from zero to MM

a m O z th t is, fro to

n a n n x nc c c all I tegr ti g e t with refere e to y, we olle t the

' c umn in KL H a n m z KL ol s the slic e N , y v ryi g fro ero to ;

a m O = b th t is, fro to y

Thi a u o f n m ua n o f cu s v l e y is take fro the eq tio the rve ALB. 27 2 INTEGRAL OAL OUL US.

F na n a nc a i lly, we i tegr te with refere e to , to c ollect all the c i n n o A sli es the e tire s lid B C. Here a: varies from zero to OA a m O ; th t is, fro to a . Hen c e we have

3 2 2 2 N1 7 ? “2 b” dcc dy dz.

E a ua n mte ral ﬁnd v l ti g g , we

7r d be V= 6

Fo r n the e tire ellipsoid,

EX A M P L ES.

Fin d the volume o f o ne o f the wedges cut from the cylin 2 2 2 x = a b an der, + y , y the pl es = = z 0 and z x tan a .

S x tan a “tan “ Ans 2 dwdy dz 3

Find the volume o f the solid c ontained between the paraboloid o f revolu tion

2 = z y a ,

2 ” n x : 2 am the cyli der + y ,

and the plan e z O.

s 2 do:d dz A n . y

Fin d the volume bounded by the surfac e

1 ,

- ' n and by the positive sides o f the three co o idinate pla es . be A ns 9 0

APT CH ER XII.

~ H Y PER B OLIC F U NCTIO NS . EQ U A TIO NS A ND P R O P ER Y P TIES O P C C L O ID, E IC Y C L O ID , A ND H Y P O C Y C L O ID .

INTR INS IC EQ U A TIO N O F A C U R V E.

e a c a c a n m c an u 68 . W h ve reserved for this h pter ert i is ell eo s ub c a m n o f c b ff n a and s je ts , for the tre t e t whi h, oth the Di ere ti l,

n a Ca cu u are u . I tegr l, l l s req ired

H ERB LIC F NCTIONS YP O U .

B ana x n n al u f e nitio ns . a o 69 . D ﬁ y logy with the e po e ti v l es n and c n o n a 60 the si e osi e, p ge ,

(30 8 33 the real function s 6 3 6 4 and 2 2

erbo li c s ine and h er bo lic co sine o f x and are called the hyp , yp , , written

sinh a c osh a: 2 2

’ By substituting a v 1 a: i n we ﬁnd

c o sh w co s (x v a n

evident also that

nh 0 Si O ,

n x n x c m. si h ) si h , osh H YP ERR OLIC FUNCTIONS. 275

The unc n sinh w c o sh a' fo r a a u o f f tio s, , , re l v l es c unc n k si n 90 c o s at but nc a periodi f tio s li e , , i re se n inﬁ ity . The other hyperbolic function s are

sinh x e” e” an a: t h ‘ c o sh a: e e

1 ei5 e" 3 c a: oth , tanh a: e” e ” 1 2 c a; se h ’ ac “ co sh a: e e

1 2 c osech x O ”c ‘ “ s1nh oc e e

From these deﬁnitions we ﬁnd

z 2 co sh x n a: 1 si h ,

ﬁ’ 2 tanh x c a: 1 se h ,

2 2 c o th a: c o sech x 1 ,

n 2 so 2 n .v c w si h si h osh ,

z z c 2 a: co sh a sinh x osh ,

n a: n wco sh co sh wsinh si h ( :l: y) si h y :t y,

c w c wo t n a: n osh ( j : y) osh osh y : si h si h y,

1 d: tanh cc tanh y

Inverse Hyp erbo lic Functi o ns .

93 n si h y,

—I n y si h x.

from _ ey e y 276 INTEGRAL CAL C UL US.

S n nc olvi g this with refere e to y,

2 90 1 V + ) .

—1 2 Hen c e sinh a; l o g (a: x/zz:

‘ l ? S m a t o o sh cc lo ac a i il rly, g ( V

I 1 l + x auh a: o t g , 2 1 _ x

— “? 1 ‘ l 1 1 1 w co th x tanh 10 g ,

2 - 1 _1 1 V 1 x 8 8 0 11 33 COSh -= 1o g a: as

- l ‘ 1 1 c osech a: sin h lo g a; a:

' ’ m D erenti atio n o H erbo lic F uncti o ns. F 72 . iﬁ f yp ro nitio ns we have

inh w o sh s c w, dx d c a: smh ac osh , Ola:

z an a: sech w t h , da:

z th w o se h w co c c , da;

a: c a: an at sech se h t h , dd

' c osech a: c osech a:coth a . 2}a:

To differentiate the inverse function

- 1 n x y si h ,

a; n we have si h y,

doc dy

278 INTEGRAL CA L C UL US .

7 4 . Circular and H erbo lic F uncti o ns a s rela te to yp , d a nd E T o o n quila tera l Hyp erbo la . o sh w the rigi of the

h erbo lic uncti o ns us yp f , let c onsider the circle

2 2 2 £8 y (1 . If we let

n u c a a a P A a d se tori l re O ,

w e have

a: a co s = a n d u d, y si ,

Hence OM w a c os

PM = y = a sin

“ “ W e shall no w show that if co s and sin in (1) “ ” are replaced by c osh and ” n n 1 a si h, the ( ) will pply to the equ ilateral hyper bola 2 2 ” :c 2 y a . ( )

Here the sect orial area P OA is u

W hence

From (2) and H YP ERB LI I 2 O C FUNCT ONS. 79

Hence

g u nce OM cc a co sh —- He z , a

2 M mh u P y a s , C7

similar expressions to

d an P OA in b a gle , the hyper ol , g tanh cc

“2 “ henc e tanh tan d; 2 w a in c c here s the ir le,

‘ 1 u tan tan d.

ns Exerci ses i n Hyp erbo li c F unctio .

- —1 4 2 n x t anh ta h . 1 1w — " l 3 1 inh cc 4 x 3 nh a} 2 . s (3 ) si .

“ 1 “ l tan h sin cc s e ch c o s cc .

—l “ 1 inh w s e c c cc . 4 . tan s osh

“ " 1 1 n 2 m 2 tan tanh cc tan si h .

“ ‘ l 1 2 tanb tan cc an sin 2 cc . 6 . t h

— ‘ 1 l 2 c osh co s a: c osh o os 2 cc .

* 1 2 co s cc c 2 cc . 8 . cosh osh 280 INTEGRAL CA L C L U US.

—l -l tan x an ac . 9 . y t h

‘ 1 tan an cc . 10 . y t h

“ 1 11 . n tan zc . y si h sec cc .

- I ‘ l tan tanh w tanh tanh zc . 13 . y V v

= 14 . sinh a w +

= 15 . c o sh cc 1 + [A

16 . 3 5

Express the equation o f the catenary

an d a n o f arc m x in lso the le gth the fro the verte ,

" un n 5 1 hyperbolic f ctio s . cc

A ns . a c and s a smh y osh d’

A Y L D EQUATION ND PROPERTIES OF THE C C OI .

i The c cu c b b a e nit o n . c 7 6 . D ﬁ y loid is the rve des ri ed y n in c cum nc f a c c as a n a a po i t the ir fere e o ir le, it rolls lo g str ight n li e . n A NP T b e a . s c c Let OX the str ight li e the ir le , with a u a a n n n P c b c c r di s , rolls lo g this li e, the poi t des ri es the y loid ' OB O .

282 INTEGRAL CA L CUL US .

Substituting these 1n we obtain

' w a d 7T a n ( ) si d,

a a c os d.

' L n d 77 d an u etti g , the gle thro gh which the circle h as ' ' m A an d m n acc n o n cc and a rolled fro , o itti g the e ts y , we h ve

’ ' cc a d a n d si ,

' a a c d y os ,

the equation o f the cycloid referred to its vertex .

m Art Tan ent and No rma l. F 1 . 7 6 w e a 7 8 . g ro ( ) , h ve

2 a 1 co s d 2 a n 3: ( ) si 5,

i = : a sin d 2 a sin co s ; gg g g

dv therefore tan <5 co t dcc

7 Henc e ; 2

B ut nc P TN an ma b P T ax si e g, the gle de y with the is f nc P T an n cu and PN o X is Eg; he e is the t ge t to the rve ,

the n ormal .

’ ‘ m 1 an d 2 o f c R a i us o Curvature. F ir 7 9 . d f ro ( ) ( ) the pre ed in a c ﬁnd g rti le, we

c osec

(30 8 8 0

0 4 a 8 m L ID 283 C YC O .

Sub u n i n x n a u o f cu a u stit ti g the e pressio for the r di s rv t re, w e have

2 % 4 - 4 n — —4 n — 2 P co t a si g a si N.

nc uc PN maki n N = PN He e if we prod e to Q, g Q , Q will n be the c entre o f curvature for the po i t P .

P u am T mak n Evo lute . c N 80 . rod e the di eter , i g an d o n NR a s am c b T c di eter des ri e the c ircle NR . his ircle

as u nc N PN. will p s thro gh Q, si e Q

r = arc P = a c NQ N ON,

are NQR OA

are A therefore QR O ON RK.

nc a n i n an u a c c o n a b He e Q is poi t eq l y l id, ge er ted y c rc N R r m K a n a n the i le Q f o lo g the str ight li e KR . ' Henc e the evo lute o f the cyc lo id OB O is c omposed ’ m -c c K n two se i y loids O a d KO . 284 I TEGRA L CA L C L S N U U .

8 1 . L en th O A rc . To ﬁn d n o f arc OP Fi g f the le gth the ( g. o f Ar ub u in t . 7 6) we s stit te

W e thus obtain a = — s = 2 a sin gdd 4 a 1 co s

' 2 7r a n arc OB 8 a . If d , we h ve for the e tire , O T u a n m o f his res lt is lso evide t fro the property the evolute, from which K B : 4 OQ K a .

A rea . To ﬁnd a a b n cu and ax 8 2 . the re etwee the rve the is f ub u in o X, we s stit te A

a 1 c o s d dcc a 1 co s d dd. y ( ) , ( )

' T u a n a a B A h s we h ve for the e tire re O O ,

2 A (1 COS dd 3 n a .

Hence this area is three times that of the generating circle .

EPICYCLOID AND HYPOCYCLOID.

i n O i lo i The u E u at o E c c . c c c 83 . q f p y d epi y loid is the rve c b b a n i n c cum nc a c c c des ri ed y poi t the ir fere e of ir le, whi h f x rolls outside o a ﬁ ed circle . ’ u c c B PS o n ﬁx c c ADA S ppose the ir le rolls the ed ir le , the ' point P des cribing the epicycloid A PA .

Let B a B : b B A B OP : . 0 , 0 , O e, t

286 I NTEGRAL CAL C UL US .

i n — u a n 3 Art . 83 c an b n b If eq tio s ( ) , we h ge i to , we have ua n o f c c the eq tio s the hypo y loid,

cc = (a

y = (a b sin

W n in c c o r c c a b n 8 5 . he , the epi y loid hypo y loid, the r tio etwee a an d b n can m na < b n ua n is give , we eli i te 5 etwee the two eq tio s, n a n a n n n n a d obt i si gle algebraic equatio betwee a: a d y. m n Fo r xa c c c a 4 b. T n e ple, o sider the hypo y loid where he n Ar 4 m ua 1 t . 8 b c eq tio s ( ) , e o e

3 3 cc : c a l; 3 ¢ a c f e geos os ¢,

§ 3 n — in in y fsi ¢ gs 3 ¢ a s qS.

g g ll nc m a as n o n a 96 . W he e y , give p ge

i l i B n a n a ius O Curva ture O E c c o . 8 6 . R d f f p y d y differe ti ti g

A r 3 t . 83 a ( ) , we h ve

a - b — 3; (a + b) Sl n Z d) Sl n

— c/> co s

(a + b)

Therefore

g = tan dx EP IC Y CL OID AND H YP OCY CL OID. 287

W henc e a + 2 b g ¢ 2 2 d y a + 2 b a + 2 b 6 . _ _ 2 ( la/3 2 b 2 b div 4 b (a + b) _a_ 4) 2 b

n in mu a f r a u cu a u ﬁnd Substituti g the for l o the r di s of rv t re, we

a 2 b 2 b a 2 b 2

a 00 c c b c m c c an d If , the epi y loid e o es the y loid,

a b 1 . a 2 b

i n i Ar nc 4 b s as n t . 7 9 . He e p g,

. a ius o u tur B n n g 8 7 R d f C rva e Of Hyp o cyclo id . y cha gi g b n in b 5 A rt . 86 w e a u o f u a u i to ( ) , h ve for the radi s c rv t re c c num ca of the hypo y loid, eri lly,

4 b (a b) gl . a —2 b 2

h O n A L en t Arc F m 2 d r . 8 a 88 . . a t 6 g f ro ( ) , we h ve

2 2 —0° 4 (a b s 1n 2 b

‘ '

nc n Fi . rt He e for the e tire loop APA ( g A . w e have

Fo r c c n o n e the hypo y loid, the le gth of loop is 288 INTE RAL G CAL CUL US.

. A rea betweeen urve a n i x i To ﬁnd a 8 9 C d F ed C rcle . the rea ' - A Fi . A r o A PA B t . b u se a na ( g it is etter to pol r c ordi tes, h mu a r d. T e , for l , A was,

' a a A PA OA an d a a will g ive the re , this , less re ’ A A b e u a a. tor O , will the req ired re

n a n tan d Differe ti ti g 1 ,

/ z we a M l ﬂ sec dd h ve d, i { CM 6

wdy 1 a c e s $ 1

A n F m 3 rt 83 a d Ar t . 86 ﬁnd ro ( ) : , we

cc d d ) y d .

Therefore

1 — co s

Hen c e

' A rea APA OA (a b) (a 2 b) co s

Subtracting the area o f the sector

' A a b A O n , W e have 2 b 2 b a W 5 3 a 2 b ' A re a APA BA = 7rb a

The c orresponding area for the hypocycloid

2 7rb (3 a 2 b) a

2 0 I 9 NTEGRAL CA L C UL US.

Fo r xam u s ﬁnd n n c u n e ple, let the i tri si eq atio of

Tak n x as 1 i n u se u a n i g the verte or g , we eq tio s r n c n o f ax o f eversi g the dire tio the is Y. W e m n acc n o itti g e ts, cc a d n ( si d) ,

= a 1 — co s y ( d) .

ff n a n ua n b a n Di ere ti ti g these eq tio s, we o t i — tam _ tan % l ibfs d

Hence <1) 3

2 2 a (2 Z co s d) 4 a c os

n s 2 d 4 a n He c e co s gd si g.

Eliminating d between (1) and we have

3 4 a sin <5,

ch n n c ua n o f c c f whi is the i tri si eq tio the y loid, re erred to its x verte .

h n uatio n o t e Evo lute. 9 2 . I tri nsi c Eq f

If we differentiate the intrinsic equation o f the curve

l u o f u u . Ca . a c a w e a e b 1 Art . 114 h v , y ( ) , Dif , the r di s rv t re,

i r i : 1

’ ’ Let P be c n o f cu a u fo r O P c O , , the e tres rv t re , , respe tively, ’ ’ and u Of OP . O P , the evol te

Let

’= ’ ’ and s O P , I T I SI E N R N C QUA TION OF TH E EVOL UTE. 291

’ ’ S nc an n O P are n ma to P i e t ge ts to or ls U ,

<1»

But from (1) consequently

Henc e

Om n acc n o n s and as n o n n itti g the e ts lo ger ecess ary, we a n n ua n o f u h ve, for the i tri sic eq tio the evol te,

F r m m n n c ua n f 9 3 . o xa f o t he c e ple, ro the i tri si eq tio ycloid = s 4 a sin ¢ f ( qt) we have

’ and f (0) 4 a .

Hence the equation o f the evolute is

s 4 a (cos (D

8 b n n a as a u o f cu a u c asin . ei g eg tive, the r di s rv t re is de re g 292 INTEGRAL CAL C UL US.

EX A M P L ES.

F n n n i c u a n o f ll n cu n i d the i tri s eq tio s the fo owi g rves, a d

o f their evolutes .

2 \ = = nd = n . ns . s a an a s a a . 1 y A t ¢, t 4>

3 ? £-£ nd s 2 . a Ans . s a 2

= — = f 3 . r a 1 co s d . Ans . s 4 a vers and s ( ) g,

294 IN TE RA L AL G C C UL US.

Since y is c onstant for points in the plane it is evi n n n f n ’ de t that the ta ge t o the a gle M P X is . the p artial

f n a c c n o f z c to cc a dif ere ti l oefﬁ ie t with respe t 3 th t is,

’ ’ 9 tan M P X = 3 . dcc

dz _ S1m1larl tan N P Y . y, ay

A s the tan gent plan e at P c ontain s the two tan gent lines ’ an M P n n n PM and PM,the pl e IW is the ta ge t pla e . ’ ’ an a a at anc h ab P ass a pl e p r llel to X Y the dist e ove it, ’ ’ n c n an n n in n M N i terse ti g the t ge t li es the poi ts , , whose

c n are M N. proje tio s , MN an d P T n cu a an d c ane Draw , perpe di l r to it, ere t the pl ’ ’ ’ P TT perpendicular to X Y .

’ n T P T The y,

’ ’ ’ ’ n n an M P the angle made by the ta ge t pl e N with X Y .

a P b. Let PM , N

By similar triangles

2 2 h h a b r V tan T P T T a b

2 2 ’PM tan ’P tan M N N;

2 8 6 0 7 1 TANGENT P LANE. 295

no ther Metho . 9 5 . A d

Let a 8 be an ma b n ma , , , y, the gles de y the or l to the ’ ’ ’ u ac at P PX P Y P Z . s rf e with , ,

Le t angles

T The c n c n o f PM are c o s A 0 sin A dire tio osi es , , ;

’ f P . o N 0 c o B in . , , s , s B

’ ’ S nc n ma n cul a b PM and P i e the or l is perpe di r to oth N , we mu a c o s a co s A c o s sin A O st h ve y , and co s c o s B c o s sin B 0 B y ,

m c c o s a tan A c o s fro whi h y,

n B ta B c OS y.

Substituting these expression s in

2 2 2 c o s a co s B 0 0 8 ‘ 1 , y ,

2 2 2 e a co tan A tan B 1 1 w h ve s 7 ( ) ,

z 2 Z sec y 1 tan A tan B 1

2 se c 7 se 2 tan A

2 2 sec 7 sec 7 2 tan B

A NA LYTIC GEO NIETRY

PLANE AND SO L D I .

B Y B W N IC H O L . . S ,

‘ ' so o Ma thema tics i n the Uir ima li Prof es r f g Mi tary Institute.

The mm o f the author has b e e n to pre p are a work fo r b e

nne and at the ame t me to make it ufﬁ c e nt c o m e gi rs, s i s i ly pr he nsive the e u e me nt o f the usua un e aduate c u e for r q ir s l d rgr o rs . Fo r the metho ds o f d e velo pme nt o f the vario us princ iple s he has d awn a e u o n e x e e nc e in the c a s o o m In the r l rg ly p his p ri l s r .

e a at o n o f the k all aut me and e n o e pr p r i wor , hors, ho for ig , wh s

n fre c o n u t o k e e a a ab e a e b e e e e . w r s w r v il l , h v ly s l d In the ﬁrst fe w c hapte rs e le m e nta ry e xample s fo llo w the dis

s io n o f e ac nc e . In the ub e ue nt c a te e t o c u s h pri ipl s s q h p rs, s s f

e a e a at nte r a t u o ut e ac c a te r and ar e xa mpl s pp r i v ls hro gh h h p , e so arrange d a s to partake b o th o f the nature o f a revie w and an

r n s e xte nsio n o f the p re c e ding p i c iple . At the e nd o f e ac h

neral e xam e n o n a mo e e xte n e a c at o n c hapte r g e pl s, i v lvi g r d d ppli i

e duc e d a re ac e fo r the be ne t o f t o f the princ iple s d , pl d ﬁ hose

n th s who may d e sire a highe r c o urs e i e ubjec t . ’ Nic ho ls s Analytic Ge ome try is in use as the re gular text in

ate numb e o f the a e c o e e and un e t e and the gre r r l rg r ll g s iv rsi i s , has prove d itself adapte d to the ne e ds o f institutio ns with the most varie d re quire me nts .

r o d cti o r th Pa es x ii 2 . Int u n i ce Clo . g 75 p ,

Pub shers Bo sto n New Yo r Ch ca o EAT 8c CO . D. C . H H , li , , k , i g NUMBE RAND ITS ALGE BRA

BY A T LEFE RE . R UR V C E. H , ‘ s u o r in Pure Ma thema tics in the n ve s it In tr ct U i r y of Texa s .

In the form o f a syllab us o f le c ture s o n the theo ry o f number and its a eb a ntro duc to to a c e at c o urs in a b a lg r , i ry oll gi e e lge r , this mo nograph p re se nts a tho ro ugh e xpo sition o f numb er as c o nc e e d and us in mat mat c s in o m c o m e ens b e b iv ed he i , f r pr h i l y t o s no t a e a t o ro u e r in the s nc It n h e lr dy h ghly v sed c ie e . is o me re psyc ho lo gic al disc ussion o f me ntal p roc esse s antec e d e nt to the mar c o nc t o f numb e r b ut the s -c o n iste nt de e m nt pri y ep , elf s v lop e o f the c o nc ept thro ugh phases undre amed o f by the man who se so le no tio n of numb er is his abstrac tion from a ﬂo c k of she e p o r o f c o n the b e n a bo kno e e nt a to pile i s, whole i g dy of wledge ss i l teac n at an ta e o f t mat c ma t mat a n r o n right hi g y s g sys e i he ic l i st uc ti . Among the unive rsities that have adop te d this wo rk are H ar a nd th un t f nn an and V n v rd a e iversi ies o Pe sylv ia irgi ia .

Cloth a es i v 2 0 P r i ce . P g , 3 . ,

THE NUMBE R SYSTE M O F ALGE BRA

Treated Theo retically a nd Histo rically

BY E R B F E PHD N Y . N . H I ,

P r of esso r of M a thema ti cs i n P r i n ceto n Un i ver s ity The theore tic al part o f this boo k is an ele me nta ry exposition f the nature o f the numb r c nc e t o f the s t e nte r and o e o p , po i iv i ge , f th o u a rt c a rm o f numb e c it the t e ih o e f r iﬁ i l fo s r, whi h, w h posi iv ” h num te m o f a a h te e c n t tut t e b e b viz. t e ne a g r, o s i e r sys lge r , , g m t the ac t o n the rat na and the a na . ive , fr i , ir io l, i gi ry The histo ric al part p resents a res ume o f the histo ry o f the f m ta a t m t c a most impo rtant p arts o ele en ry ri h e i nd algebra.

Clot/z Pa es x 1 1 . Pr i ce . g , 3 ,

E T CO Pub shers Bo sto n New o r Chica o D. C . H A H 8c , Y , g li , k

C OLLE GE A LGE BRA

BY ED A D A B R . O SE LL D W W R, . .

Pro ess o r o M a thema ti cs a nd E n i nee r i n i n R u t e rs l f f g g g Co leg e .

T o k d ne ac a e m c e and c nt c his w r is esig d for d ies, oll ges s ie iﬁ h m n c o o s . It b e n t t e e ts and the u tre tm f s h l gi s wi h ele , f ll a e nt o the e arlier parts rende rs it unnec e ssary that stude nts who use it a a e e u tu a m e e e men ta a eb a sh ll h v pr vio sly s died or l ry lg r .

Amo ng its po ints o f superio rity are the fo llo wing

m t ne o f t e atm nt c mb ne th s m c 1 . o t C ple e ss r e o i d wi i pli i y.

z A o anc e o f the ab t u e and the e ab at in t eat n the . v id s r s l or e r i g more difﬁ c ult parts o f the subj e c t.

De ﬁniteness o f tat m nt — the te and o c e s s a re 3 . s e e s ps pr se

n r gene rally fo rmula ted in plai ules .

a u c n e at o n and c ea e sentat o n o f mate a fo r 4 . C ref l o sid r i l r pr i ri l the s tude nt .

S te mat c ar an e m nt o f mat a un e ac ub e c t. 5 . ys i r g e eri l der h s j

6 Fu no te o f x anat o n d e c t o n and n o mat o n use . ll s e pl i , ir i , i f r i , ful to studen t and te ac her.

Num ro u e xam e are d tr but t u ut the text in 7 . e s pl s is i ed hro gho

r n t u t imme diate c onne c tio n with the p i c iples hey ill s rate .

i i i 0 In tr o du cti o n r i ce H a lea ther . P a es x v . lf g , + 5 4 p ,

’ ademic Al eb ra Bowser s Ac g , ’ me r Bo w ser s Plane a nd So lid Geo t y, ’ m 0 f Place and S herical Tri o no et cents . Bowser s El ements o p g ry, 9 bles W ith ta , ’ a e Lo arithmic and Tri o no me ic Tables 0 cents . Bo w ser s Five Pl c g g tr , 5 ’ a ise o n Tri ono metr Bowser s Tre t g y ,

E T 8c C Pub shers Bo sto n New Yo r Chica o C . D. H A H Q , li , , k , g C O L L E G E A L G E BRA

E. B ELL S. BY W E STER W S, ,

Professo r of Mathematics in the Massachusetts Institute

o Techn l f o ogy .

The ﬁrst e ighte en c hapte rs have b e e n arrange d with refe re nc e to the ne e ds o f tho se who wish to make a review o f that po rtio n

a c e n ua at e c o m e te as e a s o f Alge br pre di g Q dr ic s . Whil pl r g rd the t e o e t c a a t o f the ub e c t o n e no u e xam e are h r i l p r s s j , ly gh pl s

n a a e th as give n to fur ish r pid r view in e c l sro o m . Atte ntio n is invite d to the fo llowing p artic ulars o n ac c o unt o f whic h the b o o k may j ustly c laim supe rio r me rit The pro o fs o f the ﬁve fundame ntal laws o f Alge bra the C o m mutat e and A o c at e La fo r A t n and Mu t c at o n and iv ss i iv ws ddi io l ipli i , — the Distributive Law fo r Multiplic ation fo r po sitive o r negative

nte e and o t e o r ne at e ac t ns the o o s o f the i g rs, p si iv g iv fr io ; pr f fundame ntal laws o f Alge b ra fo r irratio nal numb ers the pro o f o f the Bino mial The o re m fo r po sitive i ntegral e xpo ne nts and fo r

’ frac tio nal and ne gative e xpo ne nts the pro o f o f De sc arte s s Rule o f S n fo r Po s t e Ro o t fo r nc o m e te as e as c o m e te ig s i iv s , i pl w ll pl e quations ; the Graphic al Re pre sentatio n o f Func tio ns ; the so l n a at c ua n utio n o f C ubic a d Biqu dr i Eq tio s . In Appe ndix I will b e fo und graphic al de mo nstratio ns o f the fundame ntal laws o f Alge bra fo r pure imaginary and c o mplex ’ numb e rs and in A e d x C auc ro o t at e e e ua pp n i II, hy s p f h v ry q

n ha a ro o t tio s .

Ha l leather. Pa es vi 8 . Intro duction rice f g , 5 7 p ,

Pa rt 11 be innin with ua dra tics . 1 a es . Intro ductio n rice , g g Q 3 4 p g p ,

T r D. EA 8L C O . Pu s e s Bo t n r C . b s o New Yo Ch ca o H H , li h , , k , i g TREATI SE O N TRI GO N OMETRY

A ND IT S A PPLICA T IO NS T O ASTRO NO MY AND GEO DESY

Y ED A D A. BO E B W R W S R, LL. D. P r of ess o r of M a the ma ti cs a nd E ng ineeri ng i n R utge rs Co lleg e

The aim o f the autho r has b e e n to present in as c onc ise a m as is c o ns tent t c ea rne ss the u e st c o u se in Tr for is wi h l , f ll r igo no me try whic h is give n in the best tec hnic al sc hoo ls and in a dvanc ed c o urses in c ollege s . The exam e are r nume u and are c a u e e c t pl s ve y ro s ref lly s l ed . Among these a re so me o f the most ele gant theo rems in Plan e and r no m tr The num c a ut o n o f tr n Sphe ric al T igo e y. eri l sol i ia gle s h c muc atte nt n e ac c a b e n t e at in de ta as e e . re iv d h io , h se i g r ed il ’ The c a te r o n De Mo iv re s T e m and As t n m h p s heor , ro o y, G e and P ro n e e to ntro uc e the stude nt to eod sy, olyhed s will s rv i d s e c at n o f T n me tr a e un in some o f the high r appli io s rigo o y, r r ly fo d - k Ame ric an text bo o s .

Ex ce tin o ne t is is the mo s t co m lete Ameri can Math emati cal Monthl y p g , h p T n m tr ubli s ed in America and in o int o f ex ce llenc e is su erio r Treatis e o n rigo o e y p h , p p to t at w o rk In th e met o d o f treatment arran e ment t o ra ical e x e cutio n h . h , g , yp g ph , - n w ll s ele t ed e x erc is es it has no su e rio r . The de finitio ns o f the and n ume ro us a d e c , p “ ” functio ns are given o nc e fo r a ll and need no t be restated and mo difi ed wh en o b tuse a nd reflex angles are co ns idered . In the dev elo ment o f the t eo retical art o f the s ub ect the wo rk is e s eciall p h p j , p y Fro m the innin h inte res ting and clear . beg g t e s tudent is c arried alo ng with e nthu s ias m and wit the as surance that he is mas terin the sub ect The u nus uall lar e h g j . y g d w ell-c o s en co llectio n o f ro blems are s uited to e ver re uiremen t and b an h p y q , y d nt learns to do b n so lving th e s e the stu e y do i g . ’ Th treatment o f Tri o no metric Eliminatio n De Mo ivre s T eo re m Summatio n e g , h , - mo re co m lete t an is usuall ive n in text bo o k . Serie s etc . i s s o f , , p h y g v ti n ave bee n at e red b usin the o k in th l -r These o b ser a o s h g h y g bo e c as s o o m .

a l lea ther . Pa es x i v 08 . In tr o du cti o n r i ce H f g , + 3 p ,

’ ri mi m ri l se s Fi e Place Lo a th c and T ri o no et c Tab es 0 cents . Bow r v g g , 5 ’ r s El ements o f Plane a nd S herical T ri o nometr o cents . Bowse p g y, g les W ith tab , s er’ s Plane a nd Solid Geo met Bow ry , ’ i l b ra Bo w s er s Academ c A ge , r’ s lle e Al eb a Bows e Co g g r ,

T CO Pub shers Bo sto n New Yo r Ch ca o D. C . HEA H li , , k , i g

DESC RI PTIVE GE OMETRY L ENC E A AL BY C A . D H R W O , P . D. P r of ess o r of M a the ma ti cs i n P u rdue Univers ity

The spec ial features o f this wo rk are The me thod o f develop in the ub c t b ro b ms s ste ma t c a a an e d and u g s je y p le y i lly rr g , s pple me nted by suggestions whe n nee de d ; the large numb er o f ro b e m n the m t o o f ta t n the b e ms c in p l s give ; e h d s i g pro l , whi h, c nn e c t o n t the no ta ti o n a o t d make e e r tt o i wi h d p e , s v y le ere d drawing e ntire ly self-e xplanato ry ; the intro duc tio n o f several ub c t o f c ns e ab e d e sc t e a u uc as the ax o f s je s o id r l rip iv v l e, s h is ’ ’ afﬁ n t ax n me t Pa c a and Brianc ho n s exa ns the i y, o o ry, s l s h go e arly disc ussio n o f the c on e and c ylinder o f revo lutio n and the e in e t a t m the b nn n t se sur ac e ma b e spher , ord r h fro egi i g he f s y use d as auxiliary ; the omission o f all plates exc ept a few o f a en c a ac t g eric h r er.

l th a es x i i . P r i ce 8 0 cen ts. C o . P g , 77 , GE OMETRI C AL TREATME NT OF C URVES W hich a r e i sog o na l co nj ugate to a str a ight li n e w ith r espect to a tr i a ngle

. HD Y AA SCH ATT P . B IS C J W , Assi sta nt P r ofesso r of M a thema ti cs i n the Un i vers ity of Pen nsylva n ia

The disc ussion inc ludes the hyperbola and several aspec ts o f h T e ar e o n ate u t ate the a c at n t e ellips e . hre l g f ldi g pl s ill s r ppli io o f princ iples . r i c a er Octa v o . P a es i v . P e P p . g , + 4j , C O N I C S E C TI O N S B O LAND BY RUFUS . H W ,

r K n to a M a thema ti cs i n W o mi n Semi na i s n P . P rofessor of y g y , g ,

This manual p re sents the eleme nts o f Conic Se c tio ns in a c n e m t form suited to the c apac ity o f advanc e d lasses i G o e ry.

a es i v 0 0 . P r i ce cen ts . Cloth . P g , + , 75

u ishers Bo sto n New Yo r hica o T CO . P b C D. C . HEA H , l , , k , g Sc ie n c e .

’ allar s orl o f atter A ui e to minera lo and c e mis tr . B d W d M . g d gy h y

’ enton s ui e t o General emi st r nua l fo r th e lab o rato r c e nts A ma . . B G d Ch y . y 3 5 ’ o er s Lab o rat o r anual i n i olo A n e le me ntar ui e to th e lab o rato r s tu o f B y y M B g y . y g d y dy

an ima ls an lan ts 80 c e nts . d p . ’ n n n a n a mi st r 0 c ents . o to orse a d t so s L b o rat o r anual i n C e . B y , M W y M h y 5 ’ - C ut e s si cal Lab o rat or anual . A w e ll b alanc e c o u rs e in la b o rato r s ic s t e h Phy y M d y phy ,

ui rin i ne x e ns i v e a ara tu s . Illustrate . 80 c e n ts q g p pp d .

’ C ut e s racti cal si c s . Fo r i s ch o o ls and c o lle e s h P Phy h gh g .

’ Cla rk s et o s in i cro sco D taile e sc ri ti o ns o f s ucc e s s ul me t o s . M h d M py . e d d p f h d

’ Coit s mi ca ri me i c n i nt s l A t t i s o m f a al s s 0 c e . C e t a rt s ste o . h h . W h h y y 5

’ ol o n l x men and ri i o r i s c o o ls an d c o lle e s C t s si o o : E eri tal Desc t v e F . Phy gy p p . h gh h g Illus trate d . ’ Colt on s si olo : ri e er Course Fo r e arlie r e ars in i s c o o ls . Illustrate . Phy gy B f . y h g h h d e nt go c s . ’ Colton s racti cal Z oolo Giv ar ul tu i l nimal s 60 c e nts . e s c e s to t ca a . P gy . f dy yp

’ Grabﬁ eld and B urns s e i a blem vi w an rill Pa e r 2 c ents C m c l ro s Fo r re e d . . . h P . d p 5

’ H a t s n l r llu rat t I sec A rac tic al manua fo r s tu ents and te ac e s . I st e . y ta . p d h d

’ New ell s Ex erimental emi st r A mo e rn te xt-b o o k in c e mistr fo r i s c o o ls p Ch y . d h y h gh h and c o lle e g s . ’ Ornd o rff s ab o rator anual C o ntains ire c ti o n s fo r a c o urs e o f e x e rime n ts i n O r a nic L y M . d p g ’ e mis tr r m an ms e n e mis tr o ar s c e nts . C a ran e to ac c o R e s C . . h y, g d p y h y B d 3 5

’ Pe o on it c ell and axw ell s lant Li e A lab o rato r ui e . 0 ce nts . p , M h M P f . y g d 5

’ Remsen s r ani c emi st r A n intro uc tio n to the s tu o f the c o m o un s o f c arbo n. O g Ch y . d dy p d Fo r s tu e nts o f the ure s cie nc e o r its a lic atio n to arts d p , pp . ’ - Reb ert s s St ereo C emi st r Its e ve lo me nt and re s e nt as ec ts . h y . d p p p

’ San or s Ex erimental s c olo Part I Se nsa tio n and Pe rc e tio n . f d p P y h gy . . p

’ Shaler s i r l ar s c nts F st o o k i n Ge lo t 60 ents . o e . o o . C c B gy h , B d , 45

’ S e ar s Ino r ani c C emi str De scri tiv e and ua litativ e e x e ri me ntal and in uc tiv e h p d g h y . p ; p d

le a s the s tu e nt to o b s e rv e and t ink Fo r bi s c o o ls and c o lle e s . d d h . g h g

’ S e ar s ri e er Course i n C emi str w it c ha te r o n O r anic C e mis tr . Fo r s c o o ls h p d B f h y , h g h y h

i vi n a al e ar o r le s s to th e s ub e c t an d s c o o ps li mite i n lab o ra to r ac ilitie s . 80 c e nts . g g h f y j , h d y f

l She ard s ab o rat o r N - o k lanks fo r e x e rime nts table s fo r the re ac tio n s o f p L y ote Bo . B p ;

me tallic salts Can b e u se w it an c e mis tr . o ar s . c e nts . . d h y h y B d 3 5

’ S al in s otan P x in h s tu o f lan ts . 80 c e nts . ra tic al e r is e s t e p d g B y . c e c dy p

’ - Stev ens s C emi st r Not e o o k Lab o rato r s e e ts and c o v e rs . 0 c ents . h y B . y h 5

’ V enable s S o rt Hi sto r o f mi r Fo r s tu en ts and the e n e ral re a e r. h y Ch e st y . d g d

’ alter itne and Luca i mal Li e . A lab o rato r ui e . 0 c e nts . s s An W , Wh y f y g d 5

’ un D mic s hiti n 8 si cal easurement I . Den si t H e at Li ht . and So . I I . na W g Phy M y , , gg d y , n Me th o ds o i Ph sical M e as ure me nt P s i' Ma ne tis m Ele c tri it I Pr1n i le s a d c I I. c g , y . p , hy

al P T Parts I IV i n o ne v o ume . c Laws and rin i l l c e s and a b e s . $ p , , , 3 7 5

’ i i n a Tables Pa e r . 0 c e nts . Wh t g s M t h emati cal and Phy si cal . p 5

’ ms e r 2 c e n ts . W illi a s o ern e r ra Pa . M d P t o g phy . p 5

F o r e le m e n ta ry w o r ks se e o u r l is t of c n e ho o ks i n E le m e n ta ry S ie c .

AT H Pub e Bo to n Ne w Y o rk C c a o D. C . HE lish rs , s , , hi g M a the m a tic s

’ a n s T eor o f E uati ons A t re atis e f r c ll e la rto o o e c s se s . B h y q . g ’ ow ser s Aca emi c Al ebra . F r c o n a r l B d g o s e d y s ch o o s . ’ ’ ow ser s Coll e e Al ebra . A ull tre atme nt o f e le me ntar and a v ance to ics B g g f y d d p . ' w ser s lane and Soli o me LANE b u e a rat o tr . o n s e l . c t e s . B P d G y P , d p y 7 5 ' o w ser s El ement s o f lane and S eri cal T ri ono me r 0 c ts w it tabl t . e s B P ph g y 9 h , ’ o w ser s Treati se o n lan and S eri cal Tri onometr An a v an e w o rk f r l e c o c o B P ph g y . d d and te c nic al s c o o ls le g e s h h . ’ r i e- la ari mi Tabl 0 ts se s F v ce o t c es c . B ow P L g h . 5 ’ mb er S st em In Al eb ra T e o re tic l and i t ri a Fine s u . a s o c l N y g h h .

’ r e s n re nu m r N t Frac tio nal uati n 2 G l rt s eb a s o s . T e be s : 1 . E o s N . i b e Al L o . o o g h , q ; N i a rati E u atio ns . H e r l bra Ea m r e ze n t ro u u c o A e . c nu be r o h gh Q d q ; 3 , gh g h , p d , ’ m t m Fo llo w the n uc tiv e . c s lan eo etr s i e o ts . nopki ns P e G y . d h d 7 5 ’ n s o f t he oni c Se ti n c ts How land s El eme t C c o s . 7 5 . ' f ber and i ts Al ebra . I ntro u c to r to c o lle e c o u rse s in al ebra Le evre s Num g d y g g . ’ L eo e r Exerci s es Su le me ntar w o rk fo r rill . Pe r o ze n yman s G m t y . pp y d d , ’ ra A t o ro u rill bo o k . 60 c ts . Mccurdy s Exerci se Bo o k i n Alg eb . h gh d ' n Tr o nometr Fo r c o lle e s an d te c nical sc o o ls . Mi ller s Plane a d Spheri cal i g y . g h h it s ix- la ce tables W h p , ’ l i r A t re atise fo r c o lle e co u rse s . Ni chol s Ana yt c Geo met y . g i ’ N c ol e s Calculus Dif erentia l a nd I nte ra l. h . f g ’ n In e ral al ul s Osb orne s Differenti al a d t g C c u . ’ m ebra Fo r t xt an . nd al w i n s roble s in Al . e s d re view s 0 ts Pet erson a B d P g 3 c .

’ n a ati n A b rie a nd rac tic al tre atis e . 0 c ts . Robbins s Surveyi ng a d N vi g o . f p 5 ’ a en f s Schw att s Geo metri cal Tre tm t o Curve .

’ r A lar e numbe r o f ro ble ms s s te mati call a rran e and Wald o s Descri ptive Geo met y . g p y y g d

s ti ns 80 c ts . w ith s ugg e o . ’ it r wi t o ut answ e rs . Well s s Aca demi c Ari thmeti c . W h o h ’ s l b ra Fo r s e co n ar s c o o ls . Well s s Essenti al o f A g e . d y h ’ r it o r w it o ut answ e rs . Wells s Aca demi c Al g eb a . W h h ’ eb ra Fo r s c o o ls and c o lle e s . Well s s New Hi g h er Al g . h g ’ eb ra Well s s Hi g h er Alg . ’ i l eb ra Oc tav o Well s s Uni v ers ty A g . . ’ n in w it a a lle l eb ra . a rt I I be i n u r tics . Well s s Co g e A g P , g g h q d ' me LANE c ts . o p el s s Es senti al s o f eo t r . S u c ts . W l G y P , 7 5 , 7 5 ’ e o f eometr . R ev i s ed . 1 8 LA NE c ts . c ts. Well s s El em nt s G y ( 94 ) P , 7 5 ; 7 5

’ ane and S eri cal Tri onometr . Fo r c o lle e s and te c hnic al s c o o ls Well s s New Pl ph g y g h . ’ it s ix lace table s it Ro bbins s Surv e in and Nav i ati o n W h p , W h y g g ,

’ lane and S erical 0 c ts it table s T r P . n met . Well s s Co mplet e ri g o o y . ph 9 W h ,

LA NE b o un s e a rate l c ts . P , d p y , 7 5 ’ - ari c abl 60 c ts Well s s New Six Place Lo g thmi T es . . ' - ab e 2 c ts . Wells s Four Place T l s . 5

o r A r i th m eti c: s ee o u r l i s t bo o ks i n Eleme n ta r Ma t h F of y m d i es .

EA T H 81 Y o rk C hica C . o D. H , g