Logarithmic Tables

L OGAR I TH M I C TAB L E S

P R GE OR E W I LL I A M ON E S OF. G J

ELL VE TY CORN UNI RSI .

FOURTH E I TI N D O .

To romote the detection of errors i n the tab es one do lar i be ai d p l , l w ll p

for the first notice of ever suc error. Address rof Jones at I thaca y h P . .

l flllhflll

MA C M I L L A N A N D CO.

Y m am a , N . . GE G E OR E . W JON S .

18 93 . N N C O T E T S .

Pa s -a . EXPLANATI ON OF THE TABLES

I F R - LA E LOGARI TII MS 2- 14 . OU P C 1

s A four lace ta e of o ari thms of the na tural . num ers 1 2 i t a ta e of ro p bl l g b , , w h bl p

orti ona differences in the mar i n and of the o ari t ms of the s uares c bes s uare - roots p l g , l g h q , u , q ,

- c be roots and reci rocals of the n m ers 1 2 99 . u p u b , , .

- I I . U PLA E T I G N ME T I FUN TI NS FO R C R O O R C C O ,

A four- lace ta b e of o a ri t ms of the six rinci a tri onome tric functions i t differences for p l l g h p p l g , w h mi n tes and of the en t s of area i n radia ns. u , l g h

I I I . LOGAR I TII MS NUM BERS OF ,

- m r - i A six ace ta e of o arit s of fo fi ure n mbers th a tab e of differe nces. pl bl l g h u g u , w l

NSTANTS OF MATHEMATI S AND NA — W I GHTS AND MEASU ES CO C OF TURE E R ,

A ta e of useful consta nts with the o arithms of t ose in common use . bl , l g h

ADDI TI N - SUBT A TI N LOGARI THMS O R C O ,

A six - ace ta b e of o ari thms so re ate d t a t their use the lo arithm of the sum a nd of the pl l l g l h , by , g

‘ diiterence of two numbers may be found from their logari thms wi thout ta king out the numbers

SI NES AND TANGENTS SMA ANGLES OF LL , ” ” ° ° A tab e of the ra tios si n A :A ta n A :A for an e s 0 - b ere the o ari t mic si nes and l , gl , wh by l g h

tan ents e sma are m actl t an Tab e I I g of th se ll angles found ore ex y h by l V . TRI GON MET I UNCTI NS O R C F O ,

A nve - lace ta b e of na tura sine s cosines tan ents a nd cota n enta of an es to p l l , , g , g gl

’ minutes and a six - lace ta e of t eir lo arithms i t diderences of o arithms for seconds ex , p bl h g , w h l g

ressed in units of the si th decima lace p x l p .

VI I NAT RAL A I THMS 105 - 111 I . U OG L R ,

- A six ace ta ble of na tura o arit ms of the decima num ers . 01 . 02 . (B of the na tura pl l l g h l b , , , , l

n mbers 1 2 8 1218 and of the rime num ers et een 1218 and 10000. u , , , . , p b b w

I ME AND M OSI TE NUMBERS 118 - 187 PR CO P ,

A ta e of ri me and com osite num e rs from 1 to 20000 with the ihctors of the com osite bl p p b , p

- num ers t at are not di vi si e 2 or b and ten lace o ari t ms of the rimes. b h bl by , p l g h p

S UARES 188- 180 Q ,

A ta e of the s ares of the na t ra 1 2 . 222. bl qu u l 1 , 1 - 1 140- 141 ounce, o ; 1 2 8 A tab e of the cubes of the decima numiee r . . . l l , , ,

S AR E- ROOTS 142- 145 QU ,

- m rs 1 2 and of A ta e of the s uare roots to fo r decimal laces of the na tural nu e . 299 bl q , u p , b , , ,

m rs 2 the decima nu e . 1 . l b , ,

XI I I UBE - R TS 148 - 15 1 . C OO ,

- im aces of e na t ra num rs 1 2 929 of the A ta e of the cu e roots to fo r dec a th e . bl b , u l pl , u l b , , ,

a 01 02 os m m s . . m . decimals 1 2 s and of the deci . . . , . , . , . l , , ,

XI V RE I R ALS 152- 158 . C P OC ,

A ta e of the reci rocals of the decimal n m e rs . 01 . 02 . 08 bl p u b , , ,

— UA TE - S UARES 15 1 151 Q R R Q ,

- A tab e of the a r r ar s of the na t ra numbers 1 2 8 2000. te s u e . l qu q u l , , ,

BESSEL’ S E I I ENTS CO FF C ,

’ A ta e of Besse s coemcients for second t ird fourt and fift differences for i nte r ola tion. bl l , h , h , h , p XVI I BI N MI A OEFFI I ENTS . O L C C ,

A e of inomia ooefi cients for second t ird fourth and fi ft differences for inter ola tion. tabl b l , h , , h , p

I E R S OF BSE VATI N XVI I . R OR O R O ,

A ta e of ordina te s of the ro a i lit - curve val es of ro abi it i nte ra s and ot er va ues. bl p b b y , u p b l y g l , h l

o ri t l SQ Geor e Wi iam Jones. C py gh , , by g ll ’ 2 “ H f /7" EXPLANATI ON or TH E TAB LES.

N COMMO LOGARITH MS.

F L I T FORM O A OGAR HK.

Tns mus eu m:of a number is the exponent of that power to whi ch another num

er se s r s s e . h , the ba , mu t be ai ed to give the number fir t nam d

s c on use o is 10 as s s a re incom The ba e omm ly d in computati n , and mo t number

mea surable o s of 10 s s s .of a n e p wer , a common logarithm, in general, con i t int ger,

r s c and ss c a ss . the cha acteri ti , an endle de imal, the m nti a

I f so n s of one is o of 10 a number be re lved i to two factor , which an integer p wer li s 1 and 10 of 10 is c c and the other e between , then the integer exponent the hara f ss teristi c a o is . a c e r s c is , and the log rithm of the other act r the manti a The char t i ti s u u and n a if s l man po itive if the n mber be larger than nity, eg tive it be ma ler the

ss is s os c ac s c is ca e s n a v . ti a alway p itive. A negative har teri ti indi t d by the ig bo e it

. fi 7770 10 and 7770 g , X log 0 77 z:10" 0 . 777 7 X and log .

The logarithms of all numbers expressed by the same figures in the same order have different characteristics but the same mantissa for since all such numbers may f s m 10 be got one rom another by multiplying or dividing by o e integer power of , ms ff e their logarith di er by integ rs.

i : c s a s of a u e In part cular if the de imal point tand fter the fir t figure n mb r, f s is 0 af e u s is 1 a counting rom the left, the characteri tic if t r two fig re , it if fter

e fi s is 2 and so . So if l o s e fo s si thre gure , it , on , the decima p int tand b re the fir t g nifi nt u c c s i is 1 if z l a 2 ca c is . fig re, the hara teri t c one ero fol ow the de im l point, it 0 : E . 03 4 9 . 6 g , log log log .

TAB E or mm us L S Loe a .

The s set of se b s d in f logarithm of any con cutive num er , arrange a orm convenient use s 10 u ta o s. a. a s for , con tit te a ble of l garithm Such t ble, to the ba e , need give ss s c c e s only the manti a the hara t ri tics are evident.

‘ s s of c s 12 13 In thi book there are three table ommon logarithm Table I, pp. , ,

s l r s of all e - fi ure rs c c to f c give the oga ithm thre g numbe orre t our de imal places .

I I I 2 - - . 03 7 v s i s of a ll fi ure rs to ix Table , pp , gi e the logar thm four g numbe correct s

e 114 - 33 s s. . s u s decimal plac Table IX, pp 1 , give the logarithm of all prime n mber e 20000 c ec to c b low orr t ten de imal pla ces.

All t es s are a s m of h e table arr nged upon the a e general plan, that double entry, the as of nu s a to of r a nd l t figure a mber t nding at the p the page, above the loga ithm, the s on i i other figure at the extreme left and a l ne with the logar thm .

s c a I I I w s The explanation that follow apply parti ularly to T ble but, ith light

c a s s v a s s . h nge , they may er e l o for Table I and IX

TAB LE I I I .

a s e u s s a at f of and In T ble III, the fir t thr e fig re of a number t nd the le t the page, urt r f the fo h figu e at the top the mantissa of the logarithm is ound on a. line with st f u s o u u . the fir three fig re the number, and nder the fo rth figure P A A I N AB EX L N T O OF THE T LES .

ss s ess e s a e six s the s The manti a , though endl d cimal , are c rri d to place only ixth fi is ea s is ss h f gure, being that which n re t to the true value, in error by le than al a f ix u f w . O se s s s ou s f s t o unit the fig re the la t r are alway printed in ull, but the fir t

s c s of te n or e I f appear in the fir t olumn only, and at interval , when they chang . a e oc of i b s rs s chang cur in the middle a line, warning s given y ta , and then the fir t wo e f m t figur s are rea d ro the line below . 2 f 000 0 E . 3 o ll e s f 7 79 8 4 . ss s a m 7 to g , on page the manti a numb r ro begin with

of e s f 708 0 7244 8 5 a nd 8 6 s a numb r rom to , with the change to take pl ce in the f 24 5 logarithm o 7 .

f - fi ure s f s a nd The our g number ound in the table are tabular number , their loga rithm m Th iff s f s c r s s s . e c o u are tabular logarith d eren e con e utive tab lar loga ithm , ff ces c of d ff c s s of the tabular di eren , are printed in the olumn i eren e with multiple their ts b tenth par elow them .

T F O TAKE OUT TH E LOGARI TH M O A NUMB E R.

f - fi ur e a s i s i For a our g e numb r . Take out the t bular mantis a that l e in l ne with the first three figures of the number and u nder the fourth figure the characteristic is

the exponent of that intege r power of 10 which lies next below the number .

E . . lo g , log g

of ss f r r s M e f - fi ure For a number le than ou figu e . ake the numb r a our g number

by annexing zeros and follow the rule above .

E 2 = l 8 5 3 3 2 2 l l . 0 . o o 7 7 7 g. , g g , log

l 0 2 - 8 4 3 7 o 00 70 Z . 63 . g log log .

F h m ss of s or a number of more than four figures . Take out t e ta bular anti a the fir t

f s s is ss f ss u our figure , ubtract th manti a rom the next greater tabular manti a, m ltiply

ff ce so f un ni r s as the di eren o d by the remai ng figu e , a decimal, and add the product,

as ssa of rs es . a correction, to the manti the fi t four figur E . . to out g , take log 08 3 e r s is 3 issa of 8 5 13 is . 930 . The charact i tic , and the mant log 4 00003 3 0005 1 6 is . 0 . i 00005 1 u . . u ff s . The tab lar di erence , and the prod ct of by The corrected logarithm is The work may take this form

64 33 .

The labor of multiplying is shorte ned by finding the tabular difference in the col umu of differences and adding mentally that part of this difference which lies oppo

osite f of of i h s s s x h p the fi th figure the number, a tenth that wh c lie oppo ite the i t

fi ed of w c es s te se so on . gure, a hundr th that hi h li oppo i the venth figure, and

5 1 d s t 6 is 3 1 s 4 is 20 se E . . an g , in the example above, under oppo i e oppo ite , who n 2 is 3 te is sum 3 . th part , and the 0013 8 6 13 lo . 7 So, to take out g 13 7 = 4 3 i = 3 13 3 13 . d f. l 3 8 6 . ta b. x man . log , , a nd the logarithm sought is

This process of finding logarithms of numbe rs of more than four figures is inte r pola tion by proportional parts ; it rests upon this property of logarithms :that the differences of logarithms are very nearly proportional to the differences of the i r numbers when those differences are small . I T AB E EXPLANAT ON OF HE T L S.

TO TAKE OUT A NUMBER FRO! I TS LOGARI THH .

r For a mantissa found exactly in the ta ble . Join the figu e at the top that lies above the given mantissa to the three figures upon the same line at the extreme left in the f - fi ure s so ac nu s our g number thu found, pl e the decimal point that the mber hall be the product of some number t hat lies between 1 and 10by a power of 10 whose ex

o is er s p nent the charact i tic of the logarithm .

“ i

E t lo is d m . . o g. , take out log [ g rea antilogarith

5 8 652 3 3 4 c e The mantissa . 3 lies in line with 8 and under and sin e the charact r

istic is 3 r ou i t e r s nu is 38 34 . , the e are f r n eg r figu e , and the mber

" - l 003 8 3 l . 4 o . SO, g log T ” 8 0029 o take out log i . 7

ss 78 0029 i s l 602 6 si ce te s c The manti a . l e in ine with and under , and n the charac ri ti is 1 is 602 s u . 6. , the number o ght

lo So, log g

a issa not o x ac f u - fi ure s For mant f und e tly in the table . Take out the o r g antilog rithm ul ss ss the ssa and to o of the tab ar manti a next le than given manti , it j in the

of difi erence of s two iss s t difi e e quotient the the e mant a by the abular renc . “

E . . to a out lo g , t ke g

ess t bul is 5 8 3 52 - ss . 6 s fi ure is 3 8 34 The next l a ar manti a , who e four g antilogarithm , 5 83 700 5 8 3 765 4 8 a ne rly, 5 8 3 65 2 4 8 5 8 3 65 2 113 and the number sought is nearly . “ To ta ke out log

- ess ta u r a ssa is . 78 0029 os f u fi ure is 6026 The next l b la m nti , wh e o r g antilogarithm , 78 008 9 78 0101 60 72 a ne rly, 29 60 29 72

e s is . 60268 3 and the numb r ought nearly .

" To take out log

ess .t ss is . 4 7 114 5 s u a is 295 9 The next l abular manti a , who e fo r figure antilog rithm , 4 7119 7 4 71292 5 2 :14 7 nearly, 4 5 5 2 14 5 14 7

s u is 295 93 5 0 ea and the number o ght n rly .

The labor of dividing is shorte ned by finding the ta bular difi ere nce in the column of ff s us u l s its a for s d di erence , and ing the m ltip e of tenth p rt the everal pro ucts in the

urs of vi s . s s rt s c ses be co e the di ion Thu ho ened, the whole work may in mo t a done n a is mentally, and o ly the complete ntilogarithm then written down . E . 1 f difieren . n 4 8 1 3 t o ces s o s a 4 5 is 4 f g , in dividi g by , the able h w th t tenths o 1 13 and 3 i 3 e 113 s s . , that the remainder, , nearly hundr dth of

P SSI BLE ER O RORS.

oss of as e d in s t ble is f m The p ible error any logarithm, print thi a , hal a illionth , and the possible error of any ta bular difference is a millionth but the probable error is u I f ss. s s a or if a o a be m ch le everal logarithm be dded, l g rithm multiplied by the of s be exponent a high power, the re ulting logarithm may in error by more than a A A I N AB EXPL N T O OF THE T LES.

l . s s u of n r s mi lionth In uch a ca e the fifth fig re the a tiloga ithm , the fir t got by di s on is st o the s x u is f in ou and s vi i , generally tru w rthy, i th fig re o ten d bt, the eventh i figure s rarely to be used . The possible error in the result is nearly te n times greater if the logarithm be near the end of the ta ble than if near the beginning for then a iff c i v s is s the t bular d eren e, the d i or, much maller, and an error either in it or in the fie d v has a e ct. a te a cu c s ar es s s i idend gre ter If gre r c ra y be de ired, l ger tabl mu t be u ed.

LAB - SA I NG E I ES OR V D V C .

I f the numbe r whose logarithm is sought lie nearer the la rger of two ta bular num s c c a hed s i to r a ula ss ber , the orre tion may be pp , by ubtract on, the la ger t b r manti a ; and so if e la of t a ssa s c c , a given logarithm lie n arer the rger two abular m nti , the orre tion

su e t r u . may be applied, by btraction, to the larg r abula n mber

s es o s are s of fivé st a To avoid training the ey the l garithm grouped in block , and, in e d of c s of res c oss d s c tra ing the line figu a r the page and own the column , the omputer

may guide himself by correspondences of position in the blocks.

i s c c s c 18 : e as s To div de a logarithm who e hara teri ti negative Writ down, fir t quo ur m s s i s m ha tient fig e, the nu ber of time the divi or contained t t negative multiple of s f 18 u l to or n t e t i c ter s c it el which eq a , ex larg r han, the negat ve harac i ti , carry the os r a to an ss and for ss the p itive em inder the m ti a divide the manti a of quotient.

To avoid negative characteristics :Modify the loga rithms by addi ng 10 to such s U se s ff c a s . ums c s or u s e hms har cteri tic the , di eren e , m ltiple of the modifi d logarit where the subject- matter is such tha t the general magnitude of the results cannot be

To divide a modified logarithm Add such a multiple of 10as will make the sum c d h 10 s vis v ex ee the true logarit m by time the di or di ide, and the quotient is the u o d tr e logarithm, m difie .

The arithmetical complement of a logarithm is the excess of 10over the given log arithm is of ec c a r ; it the modified logarithm the r ipro al of the number . The ithme ca l c l of a a a c f the ti omp ement t bular logarithm may be re d dire tly rom table, subtract in s u es fr 9 l s 10 g the fir t fig r of the logarithm om and the a t from .

— CON N . E GH AND MEA U RE STA TS W I TS S S .

TAB LE I v .

ab I V s c l s ts of cs of a T le give the prin ipa con tan mathemati and n ture, with the log n se I f r s c of i arithms of such of them as are in commo u . the cha acteri ti a logar thm i c s n s of be negative the modified logarithm s used . In addition to the on ta t mathe

c formulm are s w s sta s . matics, ertain ho n by which the e con nt may be determined

h is P f ss r 1s o . In c em try, ro e or Cla ke the auth rity f m E La Biirnstein a re i es s cs Pro esso o . In phy i , verett, nd lt and the pr ncipal authoriti h s s of s r has n W en there have been everal determination a con tant, eithe the range bee v s c m t or of has e n c se s em gi en, where pa e per it ed, that one them b e ho n which e ed i H the s as c es . most reliable. ere meter taken in h

” I n c s es s M asu s the m has k as the onver ion tabl for Weight and e re , eter been ta en es s al ss on h i P f sso Ro s inch , with a very m l po ible error, the aut or ty of ro e r ger

of P fesso M s of U e s oas and ro r endenhall, the uperintendent the nit d State C t and Geo s is s v the the U detic Survey . Thi value al o the legal alue of meter in nited f n B f a es . er a s o n ea u o s St t The kilogram, by the det min tion the Inter atio al ur Weight 15 43 2 5 63 9 and M asu s is 3 s. e re , grain X A A I N AB E PL N T O OF THE T LES .

AD D - ITION SUBTRACTION LOGARITHMS .

TABLE V.

Addition - subtraction logarithms (Gaussiana) a re of use in finding the logarithm of s r iff f u f the um o d erence o two n mbers directly rom their logarithms. They are

- - - n as A o a s B o s C s. a is v t know l g rithm , l garithm , and logarithm The t ble di ided in o

- 1 2- 5 3 5 4 - 8 e a ts : . 4 2 5 5 5 . thr e p r pp , ,

The formula at the botto m of any pa ge show how to use the logarithms on that for t f a : a s ac at . p ge addition, the le t for ubtr tion, the right

All cases of ca n s use of a 1 a 2 e use addition be olved by the p rt , and p rt ne d be d a 3 is s r only for subtraction p rt for ubt action only .

This ta ble is arranged on the same general pla n as Table III the A - logarithms in

a ts 1 2 c of s a B - o a ms t of p r , take the pla e number in that t ble, and the l g rith tha loga

rithms B - s a 3 the c of rs and C- o a s ; the logarithm in p rt take pla e numbe , the l g rithm tha t of logari thms

” - ‘ a s 1 2 A B so B : 1 A In p rt , , , are related that log + log .

-

Gi vnN a lo AND G E ATE THAN To I ND log , g 6, log a R R log 6, F log

r 6 s c 10 te s sum as F om log ubtra t log a and add en r the table with thi A, take

B as ar is f u f its B out , a log ithm o nd rom number ; to add log a .

Or o su lo and e ha 2 a ss . , fr m log btract g 6, if the remainder be l t n , enter the table

i s i as A a B d ad . w th thi rema nder , t ke out , and log 6

The work may ta ke this form

A [pa rt l A [pa rt 2

B B

log (a + 6) log (a + 6)

GI VEN a lo 6 AND lo GREATER TR AN lo To r 1ND log , g , g a g 6, log (a

F o lo s ct lo r m g a ubtra g 6, then

the a ss 4 B a . s as ( ) If rem inder be le than , enter the table with thi remainder and

a A as is u its a add lo and t ke out , a number fo nd from log rithm to A g 6. B [pa rt l B [part 2

A A log (a — 6) log (a — b)

(6) I f the remainder be more tha n 4 enter the table with this remainder as B

a out as is o f o its u lo and t ke C, a logarithm f und r m n mber to C add g 6. B 045 0700 B [part 3

0 0

l - — og(a b) log (a 6) E P A A I THE AB X L N T ON OF T LES.

RI G OME R N T ON T IC F U CTIONS .

In this book there are two ta bles of trigonometric functions

15 - 19 s . s e e s a ia s to for Table II, pp , give the angle in d gr e , and in r d n , five minutes

s s s five s Of ten m s for the fir t five degree and the la t degree a right angle, and to inute the rest and it gives four - place logarithms of the six principal functions of these a n ff s for n s s. gle , with di erence mi ute

V I a e 5 9 is s m a ose is c s Table , p g , a upple ent ry table wh object to make more exa t uch s as s l s computation involve very ma l angle .

V 0- 104 for two a I I . 6 s es to es s i T ble , pp , give the angl minut right angle , w th their

s s os s s c e s c r l c s si x - a natural ine , c ine , tangent , and otang nt , or ect to five p a e , and pl ce f di e n f r se s o se s fi re ces o s. logarithm the function , with cond

The explanations that follow apply particularly to Table V I I with slight changes s f r l s f V I see a 10 so o e . o . they may erve al Tabl II For exp anation Table , p ge

TAB LE VI I .

° I f the angle be less than 45 or more than the name Of the function and the m e e s f to of e s nu b r of degr e in the angle are ound at the p the pag , and the minute at ° the side Of the pa ge be low the degrees ; if the angle lie betwee n 4 5 and the n f Of es f Of ame Of the unction and the number degre are ound at the bottom the page, f f i f r s o ees . s o and the minute above the number degr _ The unction are g ven every ° r t 0 f deg ee and minu e from to and they lie in line with the minutes o the angle .

c s se v s c f c s r s the The fun tion them l e , alled the natural un tion , and their loga ithm , c f s a re e s s s s logarithmi unction , print d ide by ide, the fir t in mall type, and the other I f e as n . in larg r type being more importa t a logarithm be negative, then the modi

fied logarithm is used .

At the right of the columns Of logarithmic since and cosines and between those of logarithmic tange nts and cota ngents are printed the sixtieth parts of the differences s a ff es for s Of consecutive logarithm they are the t bular di erenc econds.

Logarithmic secants and cosecants are found by subtracting from 10 the modified

logarithms of cosines and sines.

The s do s s s a nd fu s table not di tingui h between po itive negative nction , and every function is some function of four different angles every sine is the sine Of two angles s t c s of c m s so co that are upplemen ary, and the o ine their omple ent , and with every

s a nd . ine, tangent, cotangent

° ’ ° 1965 2 is s of 1 1 20 of 168 1 . E . . 7 g , on page the decimal the ine and and the ° ' ° cosine Of 78 4 0 and of 101 and is its logarithm .

T TI N LE TO AKE OUT A FUNC ON or AN A G .

I f e s be the to Of the For an angle given in degrees and minutes. the degre at p

t s ees a e or its m page, find the minu e under the degr and t ke out the numb r, logarith ,

that lies in line with the minutes and below the name Of the function sought .

I f e s be Of find s rees a nd the degr e at the bottom the page, the minute over the deg i the function sought above ts name .

° ’ t - sin 16 10 E . . g , to ake out nat

° ’ - 8 U e 16 s 10 on . 27 4 3 . nd r and nat ine, and in line with the left, read

° ’ to t out - 107 3 4 So, ake log cot

° ' 107 - 3 4 on the f a Over and log cotangent, and in line with le t, re d A I AB EXPLAN T ON OF THE T LES .

he s s a nd s s . as t For an angle given in degree , minute , econd Take out, above, functions of the two ta bular angles between which the given angle lies and to the function of the less angle add such pa rt of the excess of the function of the grea ter f t of ss as s c s o . a ngle over that the le , the e ond are one minu e

e n The correction for seconds may be comput d a d applied mentally . With loga rithmi c functions the corrections sought are the products of the tabular differences f for secoa by the number o seconds.

I f fun of a r a f c is s the ction the gre ter angle be the g e ter unction, the corre tion po i

i if be ss f c co c o is v . t ve ; but it the le un tion, the rre ti n negati e

° ' ' t - tan 106 3 3 4 7 E . . to ou g , take nat

o ' nat- tan l o6 34

° nat - tan 106 — 3 6 x H = — 28 28

° ’ ” SO to - cot 107 3 4 25 , take out log

° - cot 10 25 : 18 3 log 7 x . 18 3 9 5 00664

TO TA KE OUT AN ANGLE M I TS UN TI N FRO F C O .

“ f I f am of f be f u func a . c The tion ound exactly in the t ble the n e , the un tion o nd at

of a s to i t s l the top the column, re d the degree at the p of the page, and the m nu e in ine

fu with the nction under the degrees.

I f the f f to of c d s the name of unction be ound at the bot m the olumn, rea the degree

f m es m f o a s . at the bottom the p ge, and the inut line with the unction over the degree

For s es two s s s c s a every ine, the table giv angle , upplement ; and whi h of them h ll

a is d c o s s SO f be t ken ma e known in pra tice by ther con ideration . with the other unc i f d tions f the signs of the unctions be disregarde .

° - 5 12 9 f E . s 7 is f 0 120 to co . 9 . g , take out nat The unction ound on page , over and i n line with and over 5 9° and in line with

“ ° lo - sin f is f u 9 1 3 1 So, to take out g The unction o nd on page , under ° and in li ne wi th and under 14 8 and in line with

" 1 ° - SO to a lo tan . 206744 f is f u 9 1 5 8 , t ke out g The unction o nd on page , over and in line with and over 121° and in line with

f not f in the t out a The unction ound exactly able . Take the two t bular functions between which the given function lies and to the smaller tabular angle add such part of sixty seconds as the difierence between the function of the less angle and the given f is a Of ff unction part the tabular di erence .

I Vith logarithmic functions the number of seconds is the quotient of this difference difieren for s by the tabular ce econds. “ - - 5 12 E . a 6 . cos . 7 g , to t ke out nat

° ” ° ' ” - cos 5 9 5 1267 60 59 9 29 nat . x § ° ° ' - cos 5 9 — 25 5 1279 — 12 120 nat . 5 0

° ' ” SO to out - sin 3 1 2 22 , take log 1

° ” ' ° ' log- sin 3 1 21 76 22 148 38

sec s c the o e is With practice the ond may be omputed mentally, and wh le angl then ea d f r d irectly rom the table. A A I N AB EXPL N T O OF THE T LES .

E AND AN N SIN S T GENTS OF SMALL A GLES .

TAB LE VL

I f n be s its s so s h 1 an a gle very mall, ine and tangent are al very mall but t e r loga rithms a nd e d n are negative very larg , and they change rapi ly and at rapidly varyi g

es . s f a re not c for use is rat Such logarithm , there ore, onvenient where interpolation cessa s s V I ne ry, and in their tead the logarithm given in Table may be used they are based on the following considerations

An angle whose bounding arc is just as long as a radius is a radian it is equal to 5 ° 17’ i f 06264 7 e . to u o s c s is 2 8 . , and the n mber e ond in an angle times the number of radians. For a small angle the number of radians in the bounding arc is a little larger than sins of s a its l s if A the the angle and a little m ller than tangent it fo low that, be a s r sse s sin A A i s s mall angle exp e d in radian , the ratio a little maller, and the ratio A : is u es os c u s tan A a little larger, than nity but both th e rati approa h nity clo er s and closer as the angle grows maller .

” I f sse s ds h sin A A is s the angle be expre d in econ , t en the ratio a little maller c of 206264 8 and ta n A A is s than the re iprocal , the ratio a little larger than thi is s ss oc . s s r s t recipr al The e ratio change ve y lowly, and hence in erpolation alway po i ° ble the ta ble gives their logarithms for every minute from 0 to

° The cosine and cota ngent Of an angle nea r 90 a re the sine and ta ngent Of the m m of of s is comple enta ry s all angle. The logarithm the cota ngent a mall angle found by subtracting the modi fied logarithm of the tangent of the angle from 10 that of the ta ngent Of an angle nea r by subtracting the modified loga ri thm of a f f m 10 the t ngent o the complementa ry small angle ro .

T T KE T TH E I NE T EN’I ‘ F M A E O A OU S OR ANG O A S ALL NGL .

Take out the logarithm that lies below the number of degrees and in line wi th the

t s e for s s and of u of minu e ; interpolat econd , add the logarithm the whole n mber seconds in the angle . Let A be the number of seconds in an angle then

” ” ” sin A : sin A :A - A lo - sin A = lo sin ( ) , and g g ( A A) + log A, ” ” ” ° - ta n A :A A ta n A ta n A A lo A . A (tan ) , and log log ( ) g

’ " l - in 0 3 0 = lo sin 63 0 630 E . o s 1 . g , g g ( ) + log ,

5 9 30. [pp. ,

° ’ ” So - ta n 3 13 4 0 = lo ta n 11620 1 1620 lo 1 1620 , log g ( ) + g ,

s ss The angle is found by a rever e proce .

“ I “ - in - ta n E . . to ta s g , ke out log and log ° ' V I I a 61 a rs s s u 1 29 From Table , p ge , it appe that the angle o ght lie between and ° ” 1° and nea rer to 1 29 a nd by the formula

” ” ” " - — - — = log sin A log (sin A :A) = log A ; log ta n A log (ta n A :A) log A

' ° ' is 29 i . e . 1 29 and the angle ,

“ " I SO to out lo - sin - ta n , take g log ° ’ ° ' ° 3 9’ 3 ° 63 The angle lies between 3 4 0 and 3 4 1 be tween 3 and [p. z 4 l 2095 3 . , ° ’ 3 ° 9' i . e 3 4 . e . 3 0 the is i . and angle , , AB 11 EXPLAN ATI ON OF THE T LES .

MINOR TABLES .

mrmr NATU AL LOGA s. vm . R

V 105 - 117 a s s s c In table III, pp . , the n tural logarithm ( ometime improperly alled s es o c a h s u s in a a Naperian, and ometim hyperb li , log rit m ) follow their n mber p r llel a c s cs a re v the s of columns. The ch ra teri ti gi en and a change in po ition the decimal point in the number changes both the mantissa and the characteristic of the logarithm .

I X RI ME AND MP SI TE NUMBE S. . P CO O R

1 8 - 13 7 s ll f s co s t e s ss a e . 1 a Tabl IX, pp , give the prime actor of mpo i e numb r le th n

20000 ha not di si 2 or 5 - c c s of t t are vi ble by , and the ten pla e ommon logarithm the

is u and to r s is s ffic to o for primes. It a do ble entry table, find p ime it u ient l ok

- ce o s of os numbers whose loga rithms are given . The ten pla l garithm all comp ite numbers whose prime factors are smaller than 20000 a re found by adding the loga 0000 l Of s u s 2 o . rithms the factor , and of prime n mber greater than by interp ation

- - - I V UARES UBES S UARE TS CUB E TS AND E I P ALS. X X . S Q , C , Q ROO , ROO R C ROC

138 - 139 s s u r s of h - fi ure s fu Table X, pp . , give the q a e all t ree g number in ll and a cha nge in the position of the decimal point in the number ma kes twice as great a

c s s i c i . hange in the quare, and in the ame d re t on

0- 14 1 of - fi x 14 v s cu s e ure u s c si u s . Table XI, pp. , gi e the be thr e g n mber corre t to fig re

14 2- 145 wo ts s e - of all - fi ure . t a s u oo s Table XII, pp , in p r , give the q ar r t three g num

- s to u ac s o a v s s r o s u s. ber fo r pl e , and by interp l tion it gi e the qua e ro t of all other n mber

146- 15 1 - f l h - fi r . e s es cu ts o al u e Table XIII, pp , in thr e part , giv the be roo t ree g f s a nd f nu s to c e o u s. mber our pla e , , by int rpolation, all other n mber

- - l XI V . 15 2 15 3 s ec oc s of h fi ure u s Tab e , pp , give the r ipr al all t ree g n mber to four d r ol n f ac s an o all o u s . pl e , , by inte p atio , ther n mber

- xv. UA TE S UARES Q R R Q .

V 154 - 15 n a X . 7 s oss u ms fi d uc T ble , pp , make it p ible, witho t logarith , to the prod t of s m e 2000 e s o su oes c su ac o . any two numb r wh e d not ex e d , by addition and btr ti n For if w a b a n t o s a 6. , be y number , then 1

ua - s u of is of The q rter q are an even number an integer, and that an odd number ends always with the fraction but as the sum and difference of any two integers are o v or ese f c s ca c c s c either b th e en both odd, th ra tion n el ea h other in the ubtra tion .

- XVI X I I . E FI I E NTS I T LATI N V CO F C FOR N ERPO O .

Let a b c d an se s a 6 c (l e s ff ces , , , , be y rie ; ” “ ‘, l , l their fir t di eren ; a b c sec difi erences so and n e of , , ,, ,, their ond , and on let be the numb r any e m T 6 and d cou c as n t r , n, between , nting T then C , , and ’ i B ss s coefi cients c+ C c + b c 6 + a b a w th e el T, l l 5 C, ( , ,) C, , 4C , ( , ,) C, , ; ffi s T 0+ c C 6 c with the binomial coe cient 1, C , l 2 , C, . ’ Of B ss s coefiicie nts is o 0 os v is v for e el C, negative through ut, and , p iti e, C, negati e 5 d f 5 u s C . an or va s . fi . Of i co c s val e of l above , C . lue below the binom al ef ient all va ues of and are v l C, C, negati e .

X I I I . ERR RS OF BSE TI N V O O RVA O .

V a 160 is e a s :The s a s o s Table X III, p ge , in thr e p rt fir t p rt give rdinate of the

- i T h o v ts a . e s s b abso pr bability cur e, and are econd part give the pro ability that the lute magnitude Of the error does not exceed the indicated fraction of the probable error . The third pa rt tabulates four fa ctors that give the probable error of a single s vat o of s v s the s ob er ion, and the pr bable error the mean of n ob er ation Multiply fir t two factors into the squa re roOt of the sums of the squa res of the discrepancies be tween the n observations and their mean or multiply the last two factors into the sum of s t es Of n the ab olu e valu the discrepa cies . - I U A A I H M . . FO R PL CE LOG R T S

4 099 I - I TEEM 13 U LO AR S. . FO R PLACE G

7 110

73 9 6

8 5 9 7 I - A AR I M . FOUR PL CE LOG TH S.

= = - l = log 1y log i w log i ar = og § 1r ‘I I - A I ME I I 15 U U . . FO R PL CE TR GONO TR C F NCT ONS

9 0° 00’

8 9° 00’

88 ° 00’

8 7° 00’

8 6° 00'

8 5 ° 00’ I I - A I M I F I U . 16 . FOU R PL CE TR GONO ETR C NCT ONS

5 ° 00’ 8 5 ° 00’

6 ° 00’ 8 4 ° 00'

7° 00’ 8 3 ° 00'

8 ° 00' 8 2° 00'

9 ° 00’ 8 1 ° 00'

10° 00' 8 0° 00'

11° 00' 79 ° 00'

12° 00' 78 ° 00'

13 ° 00' 7 7 ° 00’

14 ° 00’ 76 ° 00'

15 ° 00’ 75 ° 00'

- I I . 18 . U A I N M I UN I N FO R PL CE TR GO O ETR C F CT O S .

25 ° 00’ 65 ° 00'

26 ° 00' 64 ° 00'

27° 00’ 63 ° 00'

28 ° 00' 62° 00'

29 ° 00' 61 ° 00’

3 0° 00' 60° 00'

9 34 6

3 1° 00' 5 9 ° 00I

I 3 2° 00' 5 8 ° 00

' 3 3 ° 00' 5 7 ° 00

° ' 3 4 ° 00' 5 6 00

165 6 9 7 16 16 29 9 6 8 7 9 65 7 15 75 9 6 28

° ’ 3 5 ° 00' 5 5 00 I I U - A I N M I I N 1 . U 9 FO R PL CE TR GO O ETR C F NCT O S.

3 5 ° 00’

3 6° 00’

3 8 ° 00’

3 9 ° 00'

4 0° 00’

4 1° 00' I I I A I M M . LOG R TH S OF NU B ERS .

000000 04 3 4 08 68 13 01 1734 2166 25 98 3 029 3 4 6 1 4 75 1 5 18 1 5 609 603 8 64 66 68 94 774 8 8 174 8 600 9 026 94 5 1 98 76 *O3 OO * 07 24 * 1 14 7 * 15 70 * 1993 * 24 15 01 28 3 7 3 25 9 3 68 0 4 100 4 5 21 4 94 0 5 3 60 5 779 6 197 66 16 703 3 745 1 78 68 8 28 4 8 700 9 11 6 95 3 2 994 7 *03 6 1 *0775

02 1 18 9 1603 2016 24 28 28 4 1 3 25 2 3 664 4 4 8 6 4 8 9 6 5 3 06 65 3 3 73 5 0 775 7 8 164 8 5 7 1 8 978 9 3 8 4 9 78 9 *019 5 *0600 * 1004 * 14 08 * 18 12 * 2216 * 26 19 *3 021 03 34 24 3 8 26 4 227 4 628 5 029 5 4 3 0 5 8 3 0 6 23 0 66 29 74 26 78 25 8 223 8 6 20 9017 94 14 98 11 *0207 *0602 *0998

04 13 93 178 7 218 2 2969 3 3 62 3 75 5 4 54 0 4 93 2 5 3 23 5 7 14 6 105 68 8 5 7664 8 44 2

9 218 9606 * O3 8 0 *076 6 * 1 15 3 * 15 3 8 I "I 9 24 * 2694 05 3 078 3 84 6 4 6 13 4 9 96 5 3 78 6 14 2 65 24 6905 7 28 6 7666 8 04 6 8 4 26 8 8 05 9 18 5 9 5 63 994 2 *03 20

06 069 8 1075 14 5 2 18 29 2206 29 5 8 3 709 4 08 3 44 5 8 48 3 2 5 5 8 0 6 3 26 6699 744 3 8 18 6 8 9 28 9 298 *003 8 *04 07 * 0776 * 1 14 5 * 15 14 07 18 8 2 225 0 26 17 298 5 3 3 5 2 4 08 5 4 8 16 5 18 2 5 5 4 7 6 276 664 0 7004 73 68 7 73 1 8 094 8 4 5 7 8 8 19

07 9 18 1 9 5 4 3 9904 *0266 * 06 26 *098 7 *13 4 7 * l 7o7 *2067 * 24 26 08 278 5 3 14 4 3 5 03 4 219 4 5 76 4 9 3 4 5 29 1 5 64 7 6004

6 3 60 6 7 16 74 26 778 1 8 13 6 8 4 90 8 8 4 5 9 198 95 5 2 99 05 *0963 * 1667 * 2018 I "23 70 I "2721 *3 07 1 09 3 4 22 3 772 4 122 4 4 7 1 5 169 5 5 18 5 8 66 6 215 65 62

69 10 725 7 795 1 8 29 8 8 64 4 8 99 0 93 3 5 968 1 *0026 1003 7 1 105 9 1403 174 7 209 1 3 4 62 4 14 6 44 8 7 4 8 28 5 169 5 8 5 1 6 19 1 65 3 1 68 7 1 75 4 9 78 8 8 8 227 8 5 65 8 903 9 24 1 9 5 79 9 9 16 *025 3 1 1 05 90 0926 1263 15 99 i 9s4 2270 2605 294 0 3 609

11 3 94 3 4 27 7 4 6 1 1 4 94 4 5278 5 6 11 5 9 4 3 6276 6608 694 0 7 27 1 8 265 9 5 8 6

12 05 74 15 60 18 8 8 2216 25 4 4 28 7 1 3 198 3 8 5 2 4 178 4 5 04 4 8 3 0 5 4 8 1 5 8 06 6 13 1 64 5 6 6 78 1 7 105 74 29 775 3 8 076 8 7 22 9 04 5 9690 *0012

13 03 3 4 065 5 0977 129 8 16 19 193 9 2260 2900 3 219 3 8 5 8 4 177 4 4 9 6 5 4 5 1 5 769 608 6 73 5 4 798 7 8 3 03 8 6 18 924 9 95 64

*0194 *05 08 * 1 13 6 * 14 5 0 I "176 3 *2076 * 23 8 9 * 2702

14 3 015 3 3 27 3 95 1 4 26 3 4 5 74 4 8 8 5 5 196 5 5 07 5 8 18

14 6 128 64 3 8 6 74 8 76 76 8 6 03 8 9 11 9 219 9 8 3 5 *014 2 * 04 49 *075 6 * 1063 ll‘l 3 70 * 16 76 * 198 2 15 228 8 25 94 29 00 3 5 10 4 120 4 4 24 4 728 5 03 2 5 3 3 6 5 64 0 6 24 6 65 4 9 7 154 7 75 9 8 06 1 8 664 9 266 95 6 7 9 8 68 *0168 * 04 6 9 *0769 * 106 8 “ 16 13 68 1667 196 7 25 64 3 16 1 34 60 4 65 0 4 94 7 5 24 4 5 5 4 1 6 13 4 6726 7022 73 17 76 13 7908 8 4 9 7 8 79 2 908 6 9 3 8 0 9 6 74 99 68 17 026 2 05 5 5 08 4 8 1 14 1 17 26 2019 23 1 1 26 03 28 9 5

3 4 78 3 76 9 4 060 4 64 1 4 9 3 2 5 8 02 ' 111 H M F M E . A U LOG RIT S O N B RS .

63 8 1 6670 695 9 7 24 8 78 25 8 113 8 68 9 95 5 2 98 3 9 *0126 *04 13 *0699 *098 6 I "I 27 2 * 15 5 8

24 15 2700 298 5 ‘ 3 8 3 9 5 25 9 5 54 2 5 8 25 6 108 6674 695 6 723 9 8 084 8 64 7 8 928 9 209 94 9 0 9 771

08 9 2 1 17 1 14 5 1 2010 284 6 3 68 1 3 95 9 4 5 14 5 069 5 3 46 5 623 67 29 7005 728 1 75 5 6 78 3 2 8 107 9 206 94 8 1 9 75 5 *0029 *03 03 *05 77 * 08 5 0 * 1124 194 3 2216 24 8 8 276 1 3 03 3 3 3 05 3 5 7 7 3 84 8

4 66 3 4 934 5 74 6 6016 65 5 6 73 65 763 4 7904 8 17 3 8 44 1 8 7 10 8 9 79 9 24 7 *005 1 * 03 19 *05 8 6 *08 5 3 * 11 21 ‘ 13 8 8 * 165 4 * 19 21 27 20 298 6 3 25 2 3 5 18 3 78 3 4 04 9 4 3 14 4 5 79 5 3 73 5 902 6 166 64 3 0 6 95 7 7221

8 010 8 5 3 6 8 798 9 060 95 8 5 9 84 6 063 1 08 9 2 1 15 3 14 14 1675 193 6 2196 24 5 6 3 23 6 3 496 3 75 5 4 015 4 274 4 5 3 3 4 79 2 5 05 1 5 8 26 6084 63 4 2 6 600 68 5 8 7 1 15 73 72 763 0 84 00 8 65 7 8 9 13 9 170 94 26 9 68 2 99 3 8 ‘019 3

0960 1215 14 70 1724 1979 223 4 24 8 8 274 2 3 5 04 3 75 7 4 011 4 264 4 5 17 4 770 5 023 5 276 603 3 628 5 65 3 7 678 9 704 1 7292 75 4 4 7795 8 54 8 8 799 904 9 9 29 9 9 5 5 0 9 8 00 *005 0 *03 00 104 8 1297 15 4 6 1 795 2044 229 3 25 4 1 2790

3 5 3 4 3 78 2 4 03 0 4 277 4 5 25 4 7 72 5 019 5 26 6 6006 6 25 2 64 99 6 74 5 69 9 1 7 23 7 74 8 2 77 28 8 4 64 8 709 8 95 4 9 19 8 94 4 3 9 68 7 993 2 *0176 0908 1 15 1 13 95 163 8 18 8 1 2125 23 68 26 10 3 3 3 8 3 5 8 0 3 8 22 4 064 4 3 06 4 5 4 8 4 790 5 03 1

5 75 5 5 996 623 7 64 77 67 18 695 8 7 198 74 3 9 8 15 8 8 3 98 8 63 7 8 8 77 9 116 93 5 5 95 94 98 3 3 054 8 078 7 1025 1263 15 01 173 9 19 76 2214 2925 3 162 3 3 99 3 63 6 3 8 73 4 109 4 34 6 4 5 8 2 5 29 0 5 5 25 5 76 1 5 996 623 2 64 67 6702 69 3 7

764 1 78 75 8 1 10 8 34 4 8 5 78 8 8 12 9 04 6 9 279

7 * * * * I" * 99 8 0 0213 04 4 6. 06 79 09 12 1 144 13 77 1609 23 06 25 3 8 2770 3 001 3 23 3 3 4 64 3 696 3 9 27 4 620 4 8 5 0 5 08 1 5 3 11 5 54 2 5 77 2 6002 6 23 2 69 21 7 15 1 73 8 0 7609 78 3 8 8 067 8 296 8 5 25

9 211 94 3 9 966 7 9 8 9 5 7 0123 *03 5 1 *05 78 *08 06 14 8 8 17 15 194 2 2169 23 96 26 22 28 4 9 3 075 3 75 3 3 979 4 205 4 4 3 1 4 65 6 48 8 2 5 107 5 3 3 2 6007 6 23 2 64 5 6 668 1 6905 7 13 0 73 5 4 75 78 8 24 9 8 4 73 ' 8 69 6 8 9 20 9 14 3 93 66 95 8 9 98 12

048 0 0702 09 25 1 14 7 13 69 15 9 1 18 13 203 4 2699 29 20 3 14 1 3 3 63 3 5 84 3 8 04 4 025 4 24 6 , 4 907 5 127 5 3 4 7 5 5 6 7 5 78 7 6007 6226 64 4 6

7 104 73 23 - 75 4 2 776 1 7979 8 198 8 4 16 8 6 3 5 9 28 9 95 07 9 725 9 94 3 *016 1 *03 78 *05 9 5 *08 13 £12 I I L GAR I THMZ N UM E I O . . S OF B RS

3 0103 0 124 7 14 64 168 1 18 98 21 14 23 3 1 25 4 7 2764 298 0 3 196 3 4 12 3 628 3 84 4 405 9 44 9 1 4 706 4921 5 13 6 5 3 5 1 5 5 6 6 5 78 1 5 996 6 211 64 25 663 9 68 5 4 7068 728 2 74 96 77 10 79 24 8 13 7 8 3 5 1 8 5 64 8 778 8 99 1 9 204 94 1 7 963 0 98 4 3 I"OO5 G *0268 *04 8 1 *0693 *0906 * 11 18 * 13 3 0 * 15 4 2

3 1 175 4 19 66 2177 23 8 9 2600 28 12 3023 3 234 3 445 3 65 6 3 8 67 4 078 4 28 9 4 4 99 4 7 10 4 9 20 5 130 5 5 5 1 5 760 5 970 6 18 0 63 90 65 99 68 09 7018 7227 74 3 6 764 6 78 5 4 8 063 8 272 8 48 1 8 68 9 8 8 98 9 106 93 14 95 22 9 73 0 9 93 8 3 2 014 6 03 54 05 62 07 69 0977 118 4 13 9 1 15 98 18 05 2012

3 2 2219 24 26 263 3 28 3 9 3 25 2 345 8 3 665 4 077 4 28 2 44 8 8 4 694 5 105 5 3 10 5 5 16 5 721 5 926 6 13 1 633 6 65 4 1 6745 695 0 7 15 5 73 5 9 75 63 7767 7972 8 176 8 38 0 8 5 8 3 8 78 7 8 99 1 9 194 9 3 98 9601 98 05 *0008 *021 1 3 3 04 14 06 17 08 19 1022 1 225 14 27 163 0 18 3 2 2034 223 6

24 3 8 264 0 28 4 2 3 044 3 246 3 44 7 3 649 4 05 1 44 5 4 4 65 5 4 8 5 6 5 05 7 5 25 7 5 4 5 8 5 65 8 5 8 5 9 605 9 6260 64 60 6 660 68 60 7060 7260 74 5 9 765 9 78 5 8 8 05 8 8 25 7 8 4 5 6 8 65 6 905 4 9 25 3 965 0 9 8 4 9 I"OO4 7 ‘024 6 34 04 44 064 2 084 1 103 9 14 3 5 18 3 0 2028 2225

3 4 24 23 2620 28 17 3 014 3 212 3 4 09 3 606 3 8 02 3 999 4 196 4 3 92 4 78 5 4 9 8 1 5 178 5 3 74 5 5 70 5 766 5 962 6 15 7 63 5 3 6744 693 9 73 3 0 7720 79 15 8 1 10 8 5 00 8 694 8 8 8 9 908 3 9 278 94 72 966 6 98 60 *005 4 3 5 024 8 04 4 2 063 6 08 29 1023 1216 14 10 1603 1796 19 8 9

218 3 23 75 25 68 276 1 295 4 3 14 7 3 3 3 9 3 5 3 2 3 724 3 9 16 4 108 4 3 01 44 93 4 68 5 4 8 76 5 068 5 260 5 4 5 2 5 64 3 5 8 3 4 6026 6 217 64 08 65 99 6790 698 1 7 172 75 5 4 774 4 793 5 8 125 8 5 06 8 69 6 8 8 8 6 9076 9266 94 5 6 9 64 6 98 3 5 ‘ 0025 *0215 *04 04 *05 93 *078 3 *0972 * 1 16 1 * 13 5 0

36 1728 19 17 2105 2294 24 8 2 2671 28 5 9 3 04 8 3 23 6 3 4 24 3 8 00 3 98 8 4 176 4 3 63 4 5 5 1 4 7 3 9 4 9 26 5 113 5 48 8 5 6 75 5 8 62 604 9 6 23 6 64 23 66 10 6 79 6 698 3 7 169 73 5 6 754 2 7729 8 101 8 28 7 84 73 8 65 9 8 8 4 5 9 03 0

9216 94 01 95 8 7 9 77 2 995 8 *014 3 ‘ 03 28 *05 13 * 0698 *08 8 3

3 7 1068 125 3 14 3 7 16 22 18 06 199 1 2175 23 60 254 4 27 28 29 12 3 096 3 28 0 3 4 64 3 64 7 3 8 3 1 4 015 4 19 8 4 38 2 4 5 65 4 74 8 4 93 2 5 115 5 298 5 4 8 1 5 8 46 6 029 6212 63 94 6 5 77 6 75 9 694 2 7 124 73 06 74 8 8 7670

8 3 98 8 5 8 0 8 761 8 94 3 9 124 9 3 06 94 8 7 9 668 9 8 4 9 *003 0

3 8 0211 03 9 2 05 73 075 4 0934 1 1 15 1296 14 76 165 6 18 3 7 2017 2197 23 77 25 5 7 29 17 3 097 3 277 3 6 3 6 3 8 15 3 995 4 174 4 3 5 3 4 5 3 3 4 712 4 8 9 1 5 070 5 24 9 5 4 28 5 606 5 78 5 5 964 6 14 2 63 21 64 99 6677 68 5 6 73 90 75 68 774 6 79 23 8 101 8 279 8 45 6 8 63 4 8 8 1 1 8 9 8 9

9 166 9 3 4 3 95 20 9 698 98 75 *005 1 I"0228 *04 05 * 05 8 2 I"O75 9 39 09 35 1 11 2 128 8 14 64 164 1 18 17 2169 234 5 2697 28 73 3 04 8 3 224 3 4 00 3 75 1 3 926 4 101 4 277 4 4 5 2 4 627 4 9 77 5 15 2 5 3 26 5 501 5 6 76 6025 6 199 65 4 8 67 22 68 96 707 1 724 5 I I I LO A I TH M N M E 23 . G R S OF U B RS.

3 9 794 0 8 114 8 28 7 84 6 1 8 63 4 8 8 08 8 98 1 9 154 93 28 95 01 96 74 984 7 ‘0020 *0192 *03 65 *05 3 8 *07 1 1 ‘08 8 3 ‘ 105 6 * 1228 4 0 14 01 15 73 174 5 19 17 208 9 2261 24 3 3 2605 2777 294 9 3 121 3 292 34 64 3 635 38 07 3 978 4 14 9 4 3 20 44 92 4 663 48 34 5 005 5 176 5 5 17 5 68 8 5 8 5 8 6029 6 199 63 70

65 40 67 10 68 8 1 705 1 7221 73 9 1 75 6 1 773 1 7901 8 070 8 24 0 8 4 10 8 5 79 8 74 9 8 9 18 9 08 7 9 25 7 94 26 95 95 9764 993 3 *0102 *0271 *044 0 *0609 *0777 *094 6 * 11 14 * 128 3 * 14 5 1 4 1 1620 178 8 195 6 2124 2293 24 6 1 2629 2796 2964 3 13 2 3 300 34 67 3 63 5 3 8 03 3 9 70 4 13 7 4 3 05 4 4 72 4 63 9 4 806

4 1 49 73 5 14 0 5 3 07 54 74 5 64 1 5 8 08 5 974 6 14 1 63 08 64 74 664 1 68 07 69 73 7 13 9 73 06 74 72 763 8 78 04 79 70 8 13 5 8 3 01 8 4 67 8 63 3 8 798 8 964 9 129 9 295 94 60 9625 9 79 1 995 6 *028 6 *04 5 1 *06 16 *078 1 *094 5 * 11 10 * 1275 * 143 9 4 2 1604 1768 193 3 2097 226 1 24 26 25 90 2754 29 18 308 2

34 10 35 74 3 73 7 3 901 4 065 4 228 4 3 9 2 4 5 5 5 4 7 18 4 8 8 2 5 045 5 208 5 3 7 1 5 5 3 4 5 697 5 8 60 6023 6 18 6 634 9 65 1 1 68 3 6 6999 7 16 1 73 24 74 8 6 7648 78 1 1 7973 8 13 5 8 29 7 8 4 5 9 8 6 21 8 78 3 8 94 4 9 106 9 268 94 29 95 9 1 9 75 2 99 14 *0075 *023 6 *03 98 *05 59 *07 20 *088 1 * 104 2 * 1203

4 3 13 64 168 5 18 4 6 2007 24 8 8 264 9 28 09 29 69 3 13 0 3 290 3 4 5 0 3 6 10 3 770 3 93 0 4 090 4 24 9 4 4 09 4 5 69 4 729 4 8 8 8 5 04 8 5 207 5 5 26 5 68 5 5 844 6004 6 163 63 22 648 1 664 0 6 799 695 7 7 1 16 7 275 75 9 2 775 1 7909 8 067 8 226 8 5 4 2 8 701 8 8 5 9 9 017 9 175

9 3 3 3 94 9 1 964 8 98 06 9964 *0122 *0279 *04 3 7 *075 2 4 4 0909 1066 1224 13 8 1 1695 18 5 2 2009 2166 24 8 0 263 7 2793 295 0 3 263 3 4 19 3 5 76 38 8 9 4 04 5 4 201 4 3 5 7 4 5 13 4 669 4 8 25 4 98 1 5 293 5 44 9 5 604 5 760 5 9 15 607 1 63 8 2 6 5 3 7 6 69 2 6 84 8 7003

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95 69 96 19 9669 9719 0068 0118 0168 0218 05 66 06 16 0666 07 16 1 163 1213 15 6 1 16 11 1660 1 710

2107 215 7 2207 25 5 4 2603 265 3 2702 3 09 9 3 14 8 3 198 3 5 93 3 69 2 403 8 408 8 4 18 6

4 5 8 1 4 68 0 5 025 5 074 5 124 5 173 5 5 18 5 5 67 5 6 16 5 665 6010 605 9 6 108 6 15 7 65 01 65 5 1

699 2 704 1 7090 7 14 0 74 8 3 75 3 2 75 8 1 763 0 79 73 8 022 8 070 8 1 19 8 4 62 8 5 11 8 5 60 8 609 8 95 1 8 999 904 8 909 7

943 9 94 8 8 9 5 3 6 95 8 5 99 26 *0024 *0073 04 14 04 62 05 11 05 60 0900 094 9 099 7 104 6 13 8 6 14 3 5 14 8 3 15 3 2

18 72 1920 1969 2017 23 5 6 24 05 24 5 3 25 02 284 1 28 8 9 2938 298 6 3 3 25 3 3 73 34 21 3 4 70 38 08 3 8 5 6 3 905 3 9 5 3 ENS I I I M E H M N . . LOGARIT S OF U B RS

4 29 1 4 33 9 4 3 8 7 4 4 3 5 4 5 3 2 4 5 8 0 4 6 28 4 677 4 8 21 4 8 69 4 9 18 4 9 66 5 014 5 06 2 5 1 10 5 15 8 5 3 5 1 5 3 99 54 4 7 5 5 4 3 5 5 9 2 5 64 0 5 73 6 5 78 4 5 8 3 2 5 8 8 0 5 9 28 5 9 76 6 024 6072 6 120 6 216 6 265 63 13 63 6 1 64 09 64 5 7 65 05 65 5 3 6601

6 69 7 674 5 6793 684 0 68 8 8 693 6 6984 703 2 708 0 7 176 7224 73 20 73 68 74 16 74 64 75 12 75 5 9 765 5 7703 775 1 7799 784 7 78 94 794 2 8 03 8 8 13 4 8 18 1 8 229 8 277 8 3 25 84 21 8 4 68 8 5 16 8 6 12 8 6 5 9 8 707 8 75 5 8 8 03 8 8 5 0 8 8 98 8 94 6 8 994

908 9 9 18 5 9 23 2 9 28 0 93 28 93 75 94 23 94 7 1 95 6 6 9 6 14 966 1 9 75 7 9 8 04 98 5 2 9 900 994 7

*004 2 *0090 *013 8 *023 3 *028 0 *03 28 *03 76 * 04 23 05 18 05 66 06 13 066 1 0709 075 6 08 04 08 5 1 08 99 09 94 104 1 108 9 118 4 123 1 1279 13 26 13 74

14 69 15 16 15 63 16 1 1 165 8 1706 175 3 18 01 18 4 8 194 3 199 0 203 8 208 5 218 0 2227 2275 23 22 24 17 24 64 25 1 1 25 5 9 2606 265 3 2701 274 8 2795 28 90 29 3 7 298 5 3 03 2 3 079 3 126 3 174 3 221 3 268 3 3 63 3 4 10 3 4 5 7 3 5 04 3 5 5 2 3 5 99 3 64 6 3 693 3 74 1

3 8 3 5 3 8 8 2 3 9 29 3 977 4 024 4 07 1 4 1 18 4 16 5 4 212

4 3 07 4 3 5 4 4 4 01 444 8 4 4 95 4 5 4 2 4 5 90 4 63 7 4 6 8 4 4 778 4 8 25 4 8 7 2 4 9 19 4 9 66 5 013 5 06 1 5 108 5 15 5 5 24 9 5 29 6 5 3 4 3 5 3 90 5 4 3 7 5 4 8 4 5 5 3 1 5 5 78 5 625 5 7 19 5 76 6 5 8 13 5 8 60 5 9 07 5 9 5 4 6001 604 8 6 09 5

6 18 9 6 23 6 6 28 3 6 3 29 63 76 64 23 64 70 65 17 65 64 66 5 8 6 705 6 75 2 6 79 9 6 8 4 5 68 9 2 6 9 3 9 698 6 703 3 7 127 7 173 7220 726 7 73 14 73 6 1 74 08 74 5 4 75 01 75 95 764 2 768 8 773 5 778 2 78 29 78 75 79 22 79 69

8 06 2 8 109 8 15 6 8 203 8 24 9 8 29 6 8 3 4 3 8 3 9 0 8 4 3 6

8 5 3 0 8 5 76 8 6 23 8 670 8 7 16 8 76 3 8 8 10 8 8 5 6 8 903 8 996 9 04 3 9 09 0 9 13 6 9 18 3 9 229 9 276 93 23 9 3 69 94 63 95 09 9 5 5 6 9 6 02 9 64 9 9 6 95 9 742 9 78 9 9 8 3 5

99 28 99 75 * 0021 *0068 *01 14 *016 1 * 0207 *025 4 * 03 00 03 93 044 0 04 8 6 05 3 3 05 79 0626 06 7 2 07 19 076 5

08 5 8 09 04 09 5 1 099 7 104 4 1090 113 7 1 18 3 1 229 1 3 22 13 69 14 15 14 6 1 15 08 15 5 4 1601 164 7 169 3 1 78 6 18 3 2 18 7 9 19 25 19 7 1 2018 2064 2110 215 7

224 9 2295 23 4 2 23 8 8 24 3 4 24 8 1 25 27 25 73 26 19 27 12 275 8 28 04 28 5 1 28 9 7 294 3 29 8 9 3 08 2

3 174 3 220 3 266 3 3 13 3 3 5 9 34 05 3 4 5 1 3 4 9 7 3 5 4 3

3 63 6 3 68 2 3 7 28 3 7 74 3 8 20 3 8 6 6 3 9 13 3 9 5 9 4 005 4 09 7 4 14 3 4 18 9 4 23 5 4 28 1 4 3 27 4 3 74 4 4 20 4 4 66 4 5 5 8 4 604 4 6 5 0 4 6 96 4 74 2 4 78 8 4 8 3 4 4 8 8 0 4 926 5 018 5 064 5 1 10 5 202 5 24 8 5 294 5 3 4 0 5 3 8 6

5 4 78 5 5 24 5 5 70 5 6 16 5 66 2 5 707 5 75 3 5 799 5 8 4 5 5 9 3 7 5 9 8 3 6029 6 075 6 121 6 16 7 6 212 6 25 8 6 3 04

6 3 9 6 644 2 64 8 8 6 5 79 6 6 7 1 6 7 17 6 763

68 5 4 6 9 00 694 6 6 9 9 2 703 7 708 3 7 1 29 7 175 7220 7 3 12 73 5 8 74 03 74 4 9 74 9 5 75 4 1 758 6 76 3 2 76 78 H M F M E I I I :LOGARIT S O N U B RS.

795 2 7998 8 04 3 ° 8 08 9 8 13 5 84 09 8 4 54 8 5 00 8 5 4 6 8 5 9 1 8 8 65 8 9 11 8 95 6 9002 9 04 7 93 21 9 3 66 94 12 94 5 7 9 5 03 9 776 98 21 9 8 6 7 99 12 995 8

023 1 0276 03 22 03 67 04 12 068 5 073 0 0776 08 21 08 6 7 1 13 9 1 18 4 1229 1275 13 20 15 9 2 163 7 16 8 3 1 7 28 1773 204 5 2090 213 5 218 1 2226

24 9 7 25 4 3 25 8 8 263 3 267 8

294 9 2994 3 04 0 3 08 5 3 13 0 3 4 01 3 44 6 3 4 9 1 3 5 3 6 3 5 8 1

3 8 5 2 3 8 97 3 94 2 3 98 7 4 03 2

4 3 02 4 34 7 4 3 9 2 44 3 7 ' 4 4 8 2

4 75 2 4 79 7 4 8 4 2 4 88 7 4 93 2 5 202 5 24 7 5 29 2 5 3 3 7 5 3 8 2 5 65 1 5 69 6 5 74 1 5 78 6 5 8 3 0 6 100 6 144 6 18 9 623 4 6 279 6 5 4 8 65 93 66 3 7 6 68 2 6 727

69 96 704 0 708 5 7 13 0 7 175 744 3 74 8 8 75 3 2 75 7 7 7622 78 90 7934 79 79 8 024 8 068 8 3 3 6 8 3 8 1 8 4 25 8 4 70 8 5 14 8 78 2 8 8 26 8 8 7 1 8 9 16 8 9 60

9 227 9 27 2 93 16 93 6 1 94 05 9 672 9 7 17 9 76 1 98 06 9 8 5 0 *01 17 *016 1 I"O2O6 l"o25 0 *029 4 05 6 1 06 05 065 0 0694 073 8 1004 104 9 109 3 113 7 1 18 2

14 4 8 14 9 2 15 3 6 15 8 0 16 25 18 90 19 3 5 1979 2023 2067

23 3 3 23 77 24 21 24 65 25 09 2774 28 19 28 6 3 2907 29 5 1 3 216 3 260 3 3 04 3 3 4 8 3 3 9 2

3 65 7 3 701 3 74 5 3 78 9 3 8 3 3

4 09 7 4 14 1 4 18 5 4 229 4 273

4 5 3 7 4 5 8 1 4 625 4 669 4 7 13 4 977 5 021 5 065 5 108 5 15 2 5 4 16 5 4 6 0 5 5 04 5 5 4 7 5 5 9 1

5 8 5 4 5 8 98 5 94 2 5 9 8 6 6 03 0 6 29 3 63 3 7 63 8 0 64 24 64 68 673 1 6774 68 18 68 6 2 6 9 06 7 168 7212 725 5 7 299 73 4 3 7605 764 8 76 9 2 773 6 7779

8 04 1 8 08 5 8 129 8 172 8 216 8 4 7 7 8 5 21 8 5 64 8 608 8 65 2 8 9 13 8 9 5 6 9000 904 3 9 08 7 9 3 4 8 93 9 2 94 3 5 94 79 9 5 22

9 7 8 3 98 26 98 70 99 13 9 9 5 7 I V M H EM N N 38 . AT ATICAL CO STA TS .

' N L S N MBERS OGA I TH MS. B ER OU LI U . L R

ese num rs re efined the e uation o 2 10 lo 8 1 m 1m]? Th e a d e 9. “17 “1 8 b by q l gi c 8 , 0 g!0 e /(w r+ a w 1+ - 1 1 w’ /21 1 (2a + b) 3 +

i n i n b the o and found, success o . y f rmula = B B C 3 B 1 77 1 M lo 6 .48429 44819 0825 1 82765 11289 18 117 u u - 9 u m/ ’I t ) g"

1 N = lo 10 9914 468 -1 / g, 7 5 6 2 . 16666 6667 8749 71858 5 0 096 6 . 098809 114 7

26585 89798 28846 M B $280

. 07575 7576 nut log : 98858 4940017414 84278 51858 25811 8558 . 98726 “I Q 85485 12m 8829 1 6667 73675 90827 78455 5686 6117207184 15204 1779 43668 285 405 1953 4242 9518 1 2819 078 19 ” ” 6264 . m 7 86580. 2581 14255 26624 0917221545 88827 92815 34 27298 ” I . 1 796 5188 1 7“ 48047 60158 0874.

‘ ' ' ' ' 1 1r 1" " E ULEB S NUMBERS. 1 4 1 1 / V V”

r 1 . 7854 . 5266 . 3168 . 1018 5642 T hese numbe s we defined by the equa tion . 2 6666 2026 — c ’/9 1 1+E‘ 1 . .

6 . 9549 . 8040 a nd found ln s cce ssi on b the form la , u , y u 4 .4058

E = C B - C B + ‘9' i u u fl l v d w.“ i n — 4 5 .w66

t en E = 1 E =5 E =61 E = 1885 E r- M521 h , , ‘ , . , . , u , 6 . 6079

E =27 02765 E = 1908 60981 99915 12145 . u , u , 7 . 7092

— r - uoxs oo r 8 . 8 106 r uum L p . ' B ULBI S 001761 11177 -19 01689 80000 9 .91 19 , 7 , . p 0 1 9 8 4 6 0 7 8 9 j 88509 055 16 02790 9m 47756 2805 618 980 068 691 415 916 091 1 26585 $798 28346 668 478 8 10 147 889 094 661 491 $7” 6715 4 2 44010W359 6181§ 42887 77144 170 -1 086 616 459 978 8 66502998201 548 0 7 . 98 1 106 104 047 948 906 868 864 m22b 15644 36199 489 18 ‘ 4 97 40909 103400248 7 28644 59922 781 701 716 784 796 791 796 781 741 . 54684 $519 5 478528 1458 262 766 779 794 m 860 884 921 968 ' w8 ‘057 7 77666 48053 885 47 6 961 1 85 753 0448 0 0 167 997 991 869 606 688 606 760 3 89 9 7 8 2 01614 73295 SW 822 7 6 7 889 98 1 -097 0196 9999 «4 660 079 791 7 902029 7 7 7 9206 5 14 09863 01775 66764 0 0 719 8 94885 9101 607 5 74007 129 ” 666 m 94462 1 666 5 468 699 784 949 988 409 088 818 9 29809. 1 1 46m 57m “18 10 98648 9 4747 0088020973 72 H Y PE R BOLI C FUN TI N 78279 22686 C O S. “ 11 88905 97105 7 90018 45941 (3 106 a = l + o - o 12 88741 862226 19353 90528 m TRI GONOMETR I C UN TI NS F C O . 611111 + a ' /8 sin a: 0 OUT cos h ‘ 006 0: 1 - 1 3+ O /S ! ' tnnh (2 — 1) 1 - l ) 1+ ’ tun 1 - l ) B 0 4 1 ‘ , / cot B i z/2 1- 2 1 h ' ‘ ° ' o= 1 0— 2 - 2 — oot / 1 2 B .0 /6 1+ eec — 8 z’ 1 E ‘ 4 1 1 h , fl + .x / + i + 1 1+ - I ) R. tu b- 1 “ = — ° ‘ 5 — ' tnn x .r z/8 +z/ z/7 ' mt r u n unou c r os cn us Loe u m e o y o . = lo sin 0= io O- M 2B o 9 1 9 B - 4 1 m g", g, , [ , v + , + 1 a 17 tnh m . 06 11 . 1 x sub 17 . c b . . . tnh x 3 x . . x x x x . x . lo oos 0: — 1 B 4 1 2 gn ) , + rownns or AND nxp nnssnn 117 u nu n 125 7 22% 913 4 7818 7876 9941 s . 1788 25 78 0210 8 9978 0' 48 4818681109580 (1° 09898 85 180 2800 2947 935 4 4 4860 4868 9997 708 4 20779 9471 5 3704 8705 “ 2797 M . 0 1175221527 (P P/B " 11 5 - 3282 8715 9567 6 0000 . 0 89 917721875 1 " 1898 - 8758 4 112 9646 7 7890 7890 (010 . 0 i $5 156 ' 4225 45 15 9710 8 0000 1 0888208665 72160 1952194780 4687 4924 97m 9 6076 6076 0000 15949 94075 5148 5887 9806 10 0000 07974 97088 01 °2S265 99wo l ' a 78210 ' ( ) LEGEND RE S ALUES. V (P P/B

— l 1 =S — S /9 a " . . . 0 29888 6113 1 — ' - = S S . .fl

° ‘7 32887 11188 19 152 1 01745 82925 19948 29577 .0 262 80619 84449 18971 19786 70860 8 ° W 164 8 l ° 101407 7 561 97944 88918 si n 1 . 01745 72S4 og si n 1 ’ ’ 45 75 1 279 18 0858 4 sin 1 log si n 1 60865 8 04 s 0 00048 49 1 867 5880 W i n 1 . 0 09 101:si n 1 I V H E MI Y EN I NEE N AND PI I Y SI . C STR , G RI G, CS 3 9

HEMI STRY HY I C S CS. . P

Wei t of 1 iter of dro e n in rams D ENSITY 0? AI R gh l hy g g , . or1 i ter of air at sea level under ressure of m , p Le t t stand for the tem era t re centi rade h for the hei t l p u g , gh of the arome ter co umn e ressed in mi imeters M for the b l xp ll , Ratio of atomic ei t of ases to s ecillo ea t w gh g p h , mass in grams of one cubic centi meter of air ; then E ansion of ases ibr 1 de ree centi ra e xp g g g d , t ) kl7oo. w ' a m i no n oum. um u m vow uu or wa r“ . Al minium A 27 Man anese Mm u , l g , , Let be the tem era t re in centi rade de rees the C p u g g ; D, Antimon 8 b 120 Merc r H . y, u y, g , densi t V the vo ume of 1 ram in cu iccenti meters then y ; , l g b ; Ars enic Aa. 76 Mo bdenum Mo. , , ly , ,

. V . . Bari um Ba 187 Nic e Ni . . V , , k l, C D C D C D V ° B d h i 0 iam t , BL, N trogen, N. ,

1 9 000 26 68 03 1-1 Boron B 11 en 998 0007 14 9980 7 9 7 , . , Oxyg , 2 9997 0008 16 9916 0084 27 9660 08 41 , Bromine B r hos orus P. , P ph , , 0 0 m m 112 la ti num 8 9999 001 16 9900 0100 28 868 ad i Pt. C u , Cd . , P , , 4 0000 17 9884 0116 29 9606 0896 a m otassi um K . l 1 ci u Ca 40 . 8 C l , , P , , 9 6 9 1 9 0186 80 96 7 0426 ar e ni m Se 999 0001 8 866 7 bon C. 12 S e . C , , l u , , 6 9 1 9 6 0161 40 9286 0770 orine CL Si icon 8L 99 7 0008 9 84 Chl , , l , 7 9998 0007 20 9826 0174 60 88 19 1196 romi um Cr Si ver A . Ch , . , l , g , 8 99& 0011 21 9806 0196 60 8888 1691 o a t Co 69 Sodi m Na . C b l , u , , 9 021 0 7794 2266 r 9 9982 0018 22 788 8 7 Cu St onti um, 8 r. , 10 9976 0026 $1 9760 0240 80 7194 2887 uorine P. 19 Su ur B Fl , , lph , . , 11 9966 008-4 24 9787 0264 90 6666 8667 . Go d Am Telluri um Te. l , , , , 12 66 0046 100 6866 48 12 H dro en I L Ti n Bn 99 y g , , , . ,

I odine I . Tita ni um Ti . ELE RI I TY , , , CT C . I ron Fe 66 un te , T gs n, W ° - mil ft of son co er a t 0 e a o ms . Resistance of 1 . pp C l g l h Lead ra nium 289 U U . , Pb , , e o m a t l ga l h s. Li i um Li t , . , inc, Z u. , h Z - Electra moti ve force of a lar ce a t C. vo ts. C k ll , l Ma nesi um M . irconium. Z r g , g , Z " h ‘ Loss ste resis for wrou t iron . 002B er s er c c e by hy gh , g p y l

e r c ic ce nti me ter . ENGI NEE I NG p ub R . r a s orxou . umn m m m o. o . s . uurrs .

PRI CTI ON. Let e the velocit of i t a out cm. er sec. V b y l gh , b p Let F. B . be the coefficient of fri ction i f the od he a t rest, b y E ectrosta ti c ‘ Electrom l . . if in motion and A the an ie of re ose . M . F , g p

F B . F. M . . - ' ua nti t 1 cou om V 10l . o . 8 10 Q y , l b . / , a dr 88 ' Cl y, y , urre nt l a m ere V 10 8 - 10 C , p , / wet “47 ‘ , Potenti a 1 o t I O‘ V l , v l , / s 1 o m 10' V’ Resi tance , h , / i ron on iron or rass . 10 . 16 oi ed 06 9 44 , b , [ l a acit l ibra d V’ 10° 9 C p y , , / - on ood 4 0n d? . 26 oi ed 00 22 8 1 w , [ l Self induction 1 enr , h y , r Ma sonr , d , . 80 y y ‘ UNITS OF a n ts ru loa . on c a .80 l y, Le t L e the en th of a merc r co um n 1 mi limeter s uare - b g , , Sa nd .68 . 8i l u y l l q , ° m i r e s. at 0 C. , a nd B the resi stance n t u o Wood on stone . 18 h ,

L . B - — . on ood see eu .26 oi ed 00 14 27 w , [ l B ni cm 980 True ohm, cm . . A . U t, . . 7 uonuw s or I LABTI OI TY . ’ e al ohm . 9972 Biemen s unit . 9407 a n the f rce i n r L g , , Le t Abe the elong tio , p o tons, l the o iginal e t the cross section in s uare inches 0 the modulus of l ng h ,/ q , I ela sticit then A /j s. y ; p H dro en ram er cou om . y g , g p l b e . rams er am ere er o r co pper: g p p p h u . Brass 6600 I ron stee 12000- 18000 , , l, Silver rams er am e re er our. , g p p p h wire 6400 wro ht 120004 8000 , ug , “18 11 3 01 5 0' I n 1 4 . e r 000 Lea d 1 1 13 0 3 701 “ . Co pp , cast, 7 , 000

wi re hard drawn 8000 os hor- bronze coco The va ues i ven in this ta e are made to conform to the , , Ph p , l g bl Gun metal 4500- 0000 Tim er 000- 060 va e of the s ecific resi stm ce of co er ada ted as the stan , b , lu p pp p I ron cast sow- soon Z inc rolle 6600 dard b the American I nstitute of E ectrica En ineers at t eir , , , d, y l l g h

- i e te m er ane. A . I . E . B . vol. 7 . 848 . mee t n , S , h , , sra am m or x x xn u ns. g p b C p ° e a 0 . i G . 8 . units be B e t T be the ultimate te nacit n ounds er s uare inc Let the specific resi stanc t n . ; L . y i p p q h C C ° a ms or a wi re 1 mm . in di the resista nce a t 0 . in e oh and L the crushin load er s ua re inch for short b oc s. g g , p q , l k C l l m on B ’ R e the resistance of a i re ams ter a nd 1 . . I f b l g, , w ° a a ra tu emcient then the resis t 0 C. , and the tempe re co : ° tance at t O is found b the formu a B t= B 1 41 . . y l . ( + 0 ’ R a . . R

2918 .0871

86600 .4614 .00889 18000- 28000 660004 10000 1 6680 W 18114 . . 660004 76011) 800004 76000 1698 .0208 .«BSS r u t w o gh .

Ro e em 11000- 18000 p , h p,

2069 was .00866

Timber oak and ine 6000- 16000 40004 2000 , p , 2096 .0267 — W EIGHT “M I L LS m POUNDS P3 8 I C ron annea e 9718 . 1286 2001 006 CUB N OT. I , l d ,

' A mi ni um l w- 170 I ron cast 4186- 466 lu , . ,

- Bra ss 476 626 stee 4864 96 06114 .00072 , l,

Bric common 110 480 rou ht 416 490 12467 .1684 k , , w g ,

- - la com act 1204 86 ati num 1800 1876 9060 . 1161 0014 0086 C y , p , Pl , o er M 660 Stone 1804 70 C pp , , Earth oose 76- 90 Tim er 80- 66 , l , b , lass Wa r ure ° 18216 1680 m G , te , p , at 82 E . .

Gra ve a nd sa nd 904 20 inc 4264 60 inc ressed 6028 0710 .O(B66 l , Z , Z , p , . 4 0 I V P H Y I . S CS.

G A I TY . SPECIFIC MB A R V “. Le n t of the se on s endu um a t ei t a ve sea eve : A mi ni m 16 —9 ° C 212 It o 7 . . I ron s e c te ord. te m . . 116 g h d p l h gh b l l lu u , , , l , , ’ ‘ — - i n i nc es sin la t. 8 h l O Bism t 2048 0806 ro r m 18 . u t o d . te 1 h , / u h , w gh , p. , . ’ ‘ ° — - - i n fee t sln lat . 8 Il i O ra ss 14 8 086 B . ead 1 5 9 48 C. . 03 1 , / , 9 L , , ’ ‘ i n me ters sln lat — 8 h 10 admi m 0 400 066 Ma nesi um 2 — . 060 . 246 , / C u , g , Acce era ti on b ra vi t er second of mean solar time ar on c a rcoa 20 104 M erc r 6—86 088 l y g y, p C b , h l, . u y, . ' — ' — in fee t si n la t . 8 It l O iamon 0 121 Nic e 1 0 d d 2 . 4 97 . 1 9 , / , k l , ’ — ' mete si n ia t . 8 lo in rs Ii ra ite N . 167 P a ti n m 0 400 4 252 , / g ph , l u ,

o er 16 400 . 098 Si er 0 00 6 4 . 06 C pp , lv , HEAT. Go d 0 400 082 S ea m at con ~ . t sta nt re ss . 47 l , , p " Ener re i red to 1 m o a r from g hea t gra f te ° ' y qu w I ce - - ( , 200 . 604 Ti n 04 00 C. . Lx ° ° , , 20 to 21 er s. C. , g I ron re 0 400 118 Z inc 04 00 0983 , pu , . , . ° ° o om 20 t 21 k - m to ea t k . a r o h 1 g f w te ir g . ° ° I H - G T. at l m 68 to 00 F. 778 ft lbs. to e 1 b. of a te r fro h w , L mal B T ca ories zrrrs r 1 B riti s ther unit . . U . U o h e a r :one cand e o er Bri ti s is the i t fro m h , l l p w , h , l gh 1 a tt = 1 o r minor ories e r second a s ermace ti a nd e r in 120 ra ns a n r 1 w j ule pe second ca l p . p c l bu n g g i hou ; ca rce l is 94 ° Hea t of va oriza ti on of water a t 100 C. 687 ca ori es e r k . ca nd es near and the ni t of i t of 1890 bo i e deci p , l p g l , ly, u l gh ( ug

B . T. U . er lb . ma e is a t e ntie t of cancels a nd nea r 1 ca ndle o e r p l ) w h , ly p w . FFI X A Va w crrv r Li e ur- :i n mi es er second 1868 60 COE CI ENTS OF E P NSI ON . o 87 l p , 1 i n me ters r se n i o e co d . I f tem era t res be mea s red in centi rade de re es en ths p , p u u g g , l g k l ° ' at tem era tur t° a re re i term s e s a t 0 IN D EX F B BFRAC'I I ON FOR Ti l ! D L18 8 p e exp ssed n of th length O . the formu a 1 a t) a nd o umes the form ula ° iam n ( + , o a t 16 . o d 24Q S A co D . by l v l by l h l C , V = V 1 8 11 i t ° ( + ) . Benzi ne 16 ori te , Flu ,

I f the Fa ren eit sca e be use d t ese form a; become arbon disul. 16 Gla ss h h l , h ul C , , , ' ’ oro form 10 oc salt It a (t a (t Chl , R k , v e E t er, 16 S i n , a . h yl Wa te r 16 anad a Ba sa m 04 28 1- 49 235 , C l , ’ ’ w m v a r wa v t o - r u x u s s 132 133 073 074 a o au s a as ro a e u n m s o r u om u es . 134 191 102 100 B D D , C, , , 054 090 030 050 b b , , , , 3 07 313 170 175 b b , , , , F, G, 03 1 090 03 1 N SOU D . ’ Ve ocit of sound in fee t and i n meters er second 167 172 093 103 l y , , p ee t Mete rs e t e Fe . Met rs . 070 092 039 05 1 F . . , ° 1 I n air a t 0 . I n ead 8987 12110 144 147 030 03 1 C l , co e r 12808 8760 ater a t 8° 4718 117 144 005 090 pp , w , ass 17060 6200 stee i re 13700 4800 110 122 031 068 gl , l w ,

m ra ture s ot er t an 0° C the i i 103 117 030 030 For te e . e oc t of so nd n p h h , v l y u 230 1 ir r ssed the form a : 1 015 t 7 . 3 2 135 100 a i s exp e by ul V V( + . ) 209 270 149 153 E I I R I TI E SP C F C G AV S. 123 071 A umini m re I ron steel 1 60 09 9 091 040 05 1 l u , pu , , , wro 9- 7 ts5 commercia t 7 . 7 . 192 194 107 103 l, ugh , B ismut Lea d 013 117 025 005 h, , admi m Ma nesi m 223 233 125 130 C u , g u , ° ar on c arcoa Me rcur a t 0 C. 086 003 020 wa s C b , h l, y , diamond Nic e 290 293 100 103 , k l, as coke a ti num ca st - 216 0 g , Pl , , C ND UCTI I “F R 11RAT O VI T O . ra i te i re and foi 212 g ph , w l, Let the t ic ness of a la te of an substance be centime o er cast Si ver h k p y p C pp , , l , ters the area of one face A s ua re ce ntimeters the tem e ra e ectro tic Su r amor ho s , q , p l ly , lphu , p u , w s monoc i nic t res of the t o ihce C. and the antit o f ea t u , Q qu y h l , i n minor ca ories t a t flo s thro h the ate in t seconds Go d 19 80 r ombic l h w ug pl ; l , h , t en : fl A t 0 I ron cas t Ti n h Q ( , , , , ure i nc p , Z ,’ 802 I ce . , THE M ME TE sa u nas R O R , I ron, Le t be the tempera ture i n contlgrade degree s a nd in 00174 Paraiil ne C F . , 6 F F a re n ei t degrees , t en ( 109 Wa ter F h h h C . , ° ° ° a d for inte r ola ti on 1 1 066 . n p C F, F C

F. C . F . C . F . C . F . C . F . C . ’ "RATS Ol FUSION . 200 892 160 802 100 212 60 122 0 82 l 196 888 146 298 96 208 46 118 6 23 T Cal. B T . Ca . B . . U . . . U

- er k e r lb. er k . er . 14 p g . p p g p lb 190 874 140 284 90 194 40 104 10 — Bism t Nic e 186 866 186 276 86 186 86 96 16 6 u h , k l , admi um a ti num 49 180 866 180 266 80 176 80 86 4 C , Pl , — I ce 80 144 Si ver 176 847 126 267 76 167 26 77 - 26 18 , l , 170 888 120 248 70 168 20 68 — 22 Lea d, Tin, — — Mercur inc 166 829 116 289 66 149 16 69 86 81 y, Z , 160 820 110 280 60 140 10 60 — 40 - 40

MEI -TING POI NTS. 166 8 11 106 221 66 18 1 6 41 — 46 - 49

C. F . C . F . ° ° ESS RE AN WE I GHT. ° ° UNI TS OF FOR E , OWER , U , A umini m a bout 600 1100 I ron. cast stee 1870 2600 C P PR D l u , l,

Bismut 260 600 Lead D e ei t of 1 ram lat . 826 620 ne s i n th , h , , y w gh g

admi um 8 16 600 Merc r - 40 - 40 i n the ei t of 1 o nd a oi rd oi s C , u y, w gh p u v up ,

- o er 1064 1980 Nic e 1460 2640 Foot- o nds er min te i n 1 orse o er 39000 C pp , k l , p u p u h p w ,

Go d 1046 1910 Paralllne 64 129 i n kilo ram o me ters e r min te l , , g p u , I ce 0 82 la tinum 1776 8280 Gra i ns er cu ic inc i n 1 ram er c ic centimeter , P , p b h g p ub , I ridium 1960 8640 Si e r 964 1760 Gra ms e r s are ce ntime te r i n 1 ound er s uare inch , lv , p qu p p q ,

I ro n i a o t 1076 1970 Tin 280 446 Pounds er c bic foot in 1 i o ram er c ic meter . 062428 , p g, b u , p u k l g p ub ,

ure a o t 1600 2900 inc 412 774 Wa tts in 1 orse - ower 746 p , b u Z , h p ,

A D I I N - . UB A I N A I M V D T O S TR CT O LOG R TH S.

— — — — A z lo a B __ lo l o lo A = lo a b . g b log a , log + g g a g 6, log 6+ g ( ) AD D I TI - U B A I A I V. ON S TR CT ON LOG R THMS. 4

3 7 16

— — — A z lo lo B = lo B = lo a lo é A = 10 a b . g b log a , g a + g g g , log 6+ g ( ) 4 4 V - H . ON M ADDITION S UBTRACTI LOGARIT S.

7690

8 104

84 8 3 - B R L GA I TI I M 415 V. ADDITION SU T ACTION O R S.

3 6 13 3 54 4 3 675 3 706 4 768 3 8 00 3 9 25 3 95 7 3 98 9 4 020 4 05 2 4 08 4 4 116 4 277 4 309 4 3 4 1 4 3 74 44 07 44 3 9 4 5 7 1 4 604 4 63 7 4 6 70 4 703 4 73 7 4 770 4904 4 93 8 4 97 2 5 006 5 04 0 5 074 5 108

5 24 5 5 28 0 5 3 15 5 3 4 9 5 3 8 4 5 4 19 5 4 54 5 5 94 5 630 5 665 5 700 5 7 3 6 5 772 5 8 07 5 95 1 5 9 8 7 6023 6 05 9 609 6 6 13 2 6 169 63 16 63 5 2 63 8 9 64 27 64 64 65 01 65 3 8 668 8 6726 6 764 68 02 68 4 0 68 78 69 16

7070 7 108 7 14 7 7 18 6 7224 726 3 73 02 74 5 9 74 99 75 3 8 75 78 76 18 765 7 769 7 78 5 8 78 98 793 8 79 79 8 020 8 060 8 101 8 265 8 306 8 34 8 8 3 8 9 8 43 0 8 4 72 8 5 14 8 68 1 8 7 24 8 766 8 8 08 8 8 5 1 8 8 9 3 8 93 6

9 107 9 15 0 9 19 3 9 23 7 9 28 0 93 24 9 3 67 954 2 95 8 6 963 1 9 675 9 7 19 9 764 98 08 998 7 *003 2 ‘0077 *0123 *0168 ‘025 9 044 2 04 8 8 05 34 05 8 0 06 27 0673 0720 0907 09 54 1001 104 8 109 6 1 14 3 119 1

13 8 2 14 3 0 14 78 15 27 15 75 1624 167 2 18 68 19 17 19 66 2016 2065 21 15 2164 23 64 24 14 24 65 25 15 25 6 6 26 17 266 7 28 72 2923 29 75 3 026 3 078 3 13 0 3 18 2 3 3 90 3 44 3 3 4 95 3 5 4 8 3 6 01 3 65 4 3 707

3 920 3 974 4 028 4 08 2 4 13 6 4 190 4 24 4 44 62 4 5 17 4 5 72 4 627 4 68 2 4 73 8 4 793 5 016 5 07 2 5 128 5 18 4 5 24 1 5 29 7 5 3 54 5 5 8 2 5 6 3 9 5 696 5 75 4 5 8 12 5 8 69 5 927 6 160 6219 6277 6 3 3 6 6 3 95 64 5 4 65 13

6 75 1 68 1 1 68 71 693 1 699 1 705 1 7 1 12 73 5 5 74 16 74 77 75 3 9 7600 76 6 2 7724 7972 8 03 4 8 09 7 8 160 8 223 8 28 6 8 34 9 8 603 8 666 8 73 0 8 7 94 8 8 5 9 8 9 23 8 98 7 9 24 7 93 12 9 3 77 9 44 3 95 08 9 5 74 964 0

9905 99 72 *003 9 * 0105 *0172 *024 0 ‘03 07

05 78 064 6 07 14 078 2 08 5 1 09 20 098 8 1265 13 3 5 14 04 14 74 15 44 16 14 168 4 19 6 7 203 8 2109 218 1 225 2 23 24 23 96 268 4 27 5 7 28 3 0 2903 29 76 3 04 9 3 122

3 4 17 3 4 9 1 3 5 66 3 64 0 3 7 15 3 78 9 3 8 64 4 166 4 24 1 4 3 17 4 3 9 3 4 4 70 4 5 4 6 4 6 22 4 93 0 5 008 5 08 5 5 16 3 5 24 1 5 3 19 5 3 9 7 5 7 1 1 5 790 5 8 70 5 94 9 6028 6 108 6 18 8 65 09 65 90 66 7 1 6 75 2 6 8 3 3 69 14 6996

73 24 74 06 74 8 9 75 7 2 765 5 773 8 78 21 8 15 6 8 24 1 8 3 25 8 4 09 8 4 94 8 5 79 8 664 9006 909 2 9 179 9 265 9 3 5 1 94 3 8 95 25 98 74 996 2 I"013 8 l"0227 *03 15 *04 04 076 1 08 5 1 09 4 1 103 1 1 121 121 1 13 02

— — - — A z lo l o lo lo A o a b . g b log a , g a g 6, g 6 + l g ( ) ’ 443 D - N H M S . D N U O O V A ITIO S BTRACTI L GARIT .

15 75 1666 175 8 18 5 0 194 2 2034 2126 2219

. 24 97 25 9 1 2684 2778 28 72 2966 3 060 3 15 5 3 5 3 5 3 63 0 3 726 38 22 3 9 18 4 11 1 4401 4 498 45 96 4 694 4 792 4 8 90 4 98 8 5 08 6 54 8 3 5 5 8 2 5 68 2 5 78 2 5 98 2 608 3

63 8 6 64 8 7 65 8 9 6 69 1 6793 68 95 6997 74 09 75 13 76 17 7721 78 25 79 29 8 034 8454 8 5 60 8 666 8 772 88 78 8 98 5 9 092 9 199 95 21 9629 973 7 984 5 9954 ‘0063 *0172 06 10 08 3 0 094 1 105 2 1163 1 274 13 8 5

1721 18 3 3 194 6 205 9 2172 228 5 23 99 28 5 5 2970 3 08 5 3 200 3 3 16 34 3 1 3 54 7 3 663 4 13 0 4 24 7 4 3 65 4 4 8 3 4 601 4 7 19 4 8 3 7 5 194 5 3 14 5 4 34 5 5 5 4 5 674 5 794 5 9 15 603 6 64 00 65 22 6644 6766 68 8 9 7012 713 5 725 9

775 5 78 79 8 004 8 129 8 3 8 0 8 5 06 8 8 8 6 9 012 9 14 0 9 267 9395 95 23 9 65 1 9 779 ‘0166 ‘0296 *05 5 6 ‘068 6 ‘ 08 16 ‘ 1078 14 73 1605 1738 18 70 2003 213 6 2270 24 04 28 06 294 1 3 211 3 34 7 3 4 8 3 3 75 5

4 166 4 3 04 44 4 1 4 5 79 4 7 18 4 995 5 134 5 5 5 3 5 694 5 8 3 4 5 975 6116 6 25 7 65 4 1 6968 73 98 75 4 2 768 6 78 3 1 7976 8 4 12 8 704 8 8 5 0 8 997 9 14 4 9 291 94 3 9 98 8 3 *003 2 *018 1 *03 3 1 *04 8 0 ‘063 0 ll"0780

138 4 15 3 6 168 8 184 0 214 6 2299 24 5 3 29 15 3 070 3 225 3 3 8 0 3 5 3 6 4 004 4 4 76 4 79 1 4 95 0 5 108 5 267 5 4 27 5 5 8 6 6067 6 228 63 8 9 65 5 0 67 12 68 74 703 7 7 19 9 768 9 78 5 3 8 017 8 18 2 8 3 4 7 8 5 12 8 678 8 8 44

93 4 3 9 678 I"0014 *018 2 *03 5 1 1029 1 199 13 70 15 4 1 1712 18 8 4 205 6 2228 274 7 29 21 3 095 3 269 3 44 4 3 6 19 3 794 3 9 70 4 4 99 4 676 4 8 5 3 5 209 5 5 66 5 74 5 6 284 664 5 6 8 26 7007 7 18 9 73 7 1 75 5 3

8 103 8 28 6 8 4 70 8 8 4 0 9 025 9 210 9 3 96 * 014 3 *03 3 1 *05 19 *0707 ll“08 96 ‘ 1274 1844 203 5 2226 24 18 26 10 28 02 2995 3 18 8 3 768 3 96 3 4 54 8 4 74 4 4 940 5 13 7

5 728 5 9 26 ‘ 6 125 65 23 6722 69 22 7 122

7725 79 27 8 129 8 3 3 1 8 5 34 8 73 7 8 94 1 9 14 5 975 9 9964 ‘0170 *03 76 ‘05 8 3 *0790 ‘099 7 * 1205 183 0 203 9 224 9 24 5 9 2669 28 8 0 3 09 1 3 3 02 3939 4 15 2 4 3 66 4 5 79 4 794 5 008 54 3 8 608 7 63 04 6 73 9 695 7 7 175 73 94 76 14

8 274 8 4 95 8 7 16 8 9 3 8 9 160 93 8 2 98 28 07 25 095 0 1176 14 02 1629 208 3 2767 2996 34 5 5 3 68 5 3 9 15 4 146 4 3 78 5 074 5 3 07 5 54 0 5 774 6008 6 24 3 64 78 6714 74 22 76 60 8 3 7 3 9 09 1

A z lo — lo l B = lo = 10 — = — o B a A 10 . g 6 g a , g a + g g log 6, log b+ g (a b) - SU B R O A I M V . ADDITION T ACTION L G R TH S.

9 5 0 95 7 2 98 12 *005 4 I "O295 *05 3 8 *078 0 * 1023 * 1 267 II"15 10 1999 2244 24 90 273 6 298 2 3 229 3 4 76 3 724 3 9 7 2 5 2 4 221 4 4 70 4 7 19 4 969 5 219 5 4 70 5 721 5 973 6 225 64 77 5 3 6 73 0 69 8 3 723 7 749 1 774 6 8 001 8 25 6 8 5 12 8 769 9025 5 4 9 28 3 95 4 0 9 799 7 005 7 *03 16 *05 76 ‘08 3 5 * 1096 * l 3 5 7 * 16 18

5 5 214 2 24 04 266 7 29 3 1 3 195 34 5 9 3 7 24 3 98 9 4 25 5 5 6 4 5 21 4 78 7 5 05 4 5 3 22 5 5 90 5 8 5 8 6 127 63 96 666 6 69 3 6 5 7 7207 74 78 775 0 8 022 8 294 8 5 6 7 8 8 4 1 9 1 14 93 8 9 9 66 3 5 8 993 9 7 0214 7 04 9 1 *076 7 7 104 4 * 1 3 22 * 1600 * 18 78 7 215 7 7 24 3 7 5 9 27 16 299 7 3 277 3 5 5 9 3 84 0 4 123 4 4 05 4 68 8 4 9 72 5 25 6

5 5 4 0 5 8 25 6 1 1 1 63 9 7 6 68 3 69 70 725 7 75 4 5 78 3 3 8 122 6 1 8 4 1 1 8 701 8 99 1 9 28 2 95 73 98 6 5 7 015 7 *044 9 7 074 2 * 103 6

6 2 0 15 1 3 3 0 16 24 19 19 2215 25 1 1 28 07 3 104 34 01 3 699 3 99 7 63 4 296 4 5 9 5 4 8 95 5 19 5 5 4 96 5 797 6099 64 01 6704 7007 64 73 10 76 15 79 19 8 224 8 5 3 0 8 8 3 6 9 14 2 94 4 9 9 75 7 *0065

65 03 74 068 3 099 2 13 02 16 13 1924 223 5 25 4 7 28 5 9 3 172 66 3 4 8 6 3 8 00 4 1 14 4 4 29 4 74 5 5 06 1 5 3 77 5 694 6 011 63 29 6 7 664 8 6967 7 28 6 7606 79 26 8 24 7 8 5 69 8 8 9 1 9 213 95 3 6 68 9 8 6 0 *018 3 *05 08 *08 3 3 * 1 15 8 * 14 8 4 * 18 11 * 213 8 * 24 65 7 2793 69 3 122 34 5 1 3 78 0 4 1 10 4 44 1 4 77 2 5 104 5 4 3 6 5 768 6 101

11 70 0J 7’ 64 3 5 6 769 7 104 74 3 9 77 74 8 1 11 8 4 4 7 8 784 9 122 94 60 71 9 799 *013 8 *04 78 *08 18 7 1 15 9 * 15 01 * 18 4 2 * 218 5 *25 28 *28 7 1 72 3 215 3 5 5 9 3 904 4 25 0 4 5 9 6 4 94 2 5 28 9 5 63 7 5 985 63 3 4 73 6 68 3 703 3 73 8 3 773 3 8 08 5 8 4 3 6 8 78 9 9 14 1 94 95 98 4 9 74 0203 05 5 8 09 13 1269 16 26 198 3 234 0 269 9 3 05 7 3 4 16

75 3 776 4 13 6 4 4 97 4 8 5 8 5 220 5 5 8 2 5 94 5 63 08 6672 703 7 76 74 02 776 7 8 13 3 8 5 00 8 8 6 7 9 23 5 9 603 99 72 7 03 4 1 7 07 1 1 77 14 5 2 18 23 219 5 25 68 294 1 3 3 15 3 68 9 4 063 4 4 3 8 78 4 8 14 5 190 5 5 67 5 94 5 63 23 6701 708 0 74 5 9 78 39 8 220 79 8 601 8 98 3 93 65 974 8 7 013 1 7 05 15 *0900 * 1284 *1670 * 205 6

021 24 4 3 28 3 0 3 217 3 606 3 994 4 38 4 4 774 5 164 5 5 5 5 5 94 7 8 1 6 3 3 9 673 1 7 124 75 18 79 12 8 3 07 8 703 9098 94 95 98 9 2 8 2 028 9 068 8 108 6 14 8 5 18 8 5 228 6 26 8 6 3 08 8 34 90 3 8 9 2 8 3 4 29 6 4 699 5 103 5 5 08 5 9 13 63 19 6726 7 13 3 754 0 794 8 8 4 8 3 5 7 8 766 9 176 9 5 8 6 999 7 7 04 09 *08 21 * 123 3 * 16 4 6 * 2060

8 5 24 74 28 8 9 3 3 04 3 720 4 13 7 4 5 5 4 4 97 1 5 3 8 9 5 8 08 6 227 8 6 664 7 706 7 74 8 8 79 10 8 3 3 2 8 75 5 9 178 9 602 *0026 7 04 5 1 8 7 024 08 76 13 02 17 29 215 6 25 8 4 3 012 34 4 1 3 8 70 4 3 00 4 73 0 8 8 5 16 2 5 5 93 6025 64 5 8 68 9 1 73 25 7760 8 195 8 6 3 0 9 067 8 9 95 03 994 1 7 03 79 *08 17 * 125 6 * 1696 * 2l 3 6 * 25 76 *3 018 7 34 5 9

3 902 4 3 45 4 78 8 5 23 3 5 6 77 6 122 6 5 68 7015 74 6 2 7909 9 1 8 3 5 7 8 8 06 9 25 5 9 705 *015 5 *0606 * 105 8 7 15 10 * 1962 * 24 16

9 2 28 69 3 3 24 3 779 4 23 4 4 69 0 5 14 7 5 6 04 606 2 65 20 6979 93 74 39 78 99 8 3 60 8 8 21 9 28 3 9 74 5 *0208 *06 7 1 7 1 13 5 *1600 94 2065 25 3 1 299 8 34 64 3 93 2 44 00 4 8 69 5 3 3 8 5 8 08 6 278

95 674 9 7221 769 3 8 165 8 63 9 9 1 13 95 8 7 *0062 *05 3 8 7 1014 96 1968 24 4 5 29 24 3 4 03 3 8 8 2 4 3 63 4 8 4 3 5 3 25 5 8 07 97 628 9 6772 725 6 774 0 8 225 8 7 10 9 19 6 9 68 2 *0169 *06 5 7 98 029 1 14 5 1634 2123 2613 3 104 3 5 95 4 08 6 4 5 79 5 07 1 5 5 65 99 605 9 65 5 3 704 8 75 44 8 04 0 8 5 3 7 9 03 5 9 5 3 3 7 003 1 7 05 3 0

— — - z l l B = :_ — A o a o lo l o a lo A l a . g b log , g a + g (ax g log b, g b+ og ( b) 4 8 AD D N - U N A I H M V. ITIO S BTRACTIO LOG R T S.

108 0 1 13 0 1 18 0 1 23 0 128 0 13 3 0 13 8 0 14 3 0 14 8 0 15 8 0 163 0 168 0 173 1 178 1 18 3 1 18 8 1 193 1 198 1 208 1 213 1 218 2 223 2 228 2 23 3 2 23 8 2 24 3 2 248 2 263 3 273 3 2784 28 34 28 84 2934 2984 3 08 5 3 13 5 3 18 5 3 23 6 3 28 6 3 3 3 6 3 38 6 3 4 3 7 348 7

3 63 8 3 68 8 3 738 3 78 9 3 8 3 9 38 89 3 94 0 3990 4 09 1 4 14 1 4 19 1 4 24 2 4 29 2 4 3 4 3 4 393 4 4 4 3 44 94 4 5 95 4 74 6 4 796 4 8 4 7 4 8 9 7 4 94 8 4 998 5 099 5 14 9 5 200 5 25 0 5 3 01 5 3 5 1 5 4 02 5 4 5 2 5 5 03 5 604 5 75 5 5 8 06 5 8 5 7 5 907 5 95 8 6008

6 109 6 160 6 211 6 26 1 6 3 12 63 62 64 13 64 64 65 14 66 15 6666 67 17 676 7 68 18 68 69 6 9 19 69 70 7021

7 122 7 173 7224 7274 73 25 73 76 74 26 \ 74 77 75 28 7629 768 0 773 1 778 2 78 3 2 78 8 3 793 4 798 5 8 03 6 8 13 7 8 18 8 8 23 9 8 29 0 8 3 4 1 8 3 9 1 8 44 2 8 4 93 8 5 4 4

8 64 6 8 696 8 74 7 8 798 8 8 4 9 8 900 8 95 1 9002 905 3 9 15 5 9 206 925 6 93 07 93 5 8 94 09 94 60 95 11 9 5 62 9664 9715 9 76 6 98 17 9 8 68 99 19 9970 *0072 0174 0225 0276 03 27 03 78 04 3 0 04 8 1 05 3 2 05 8 3 068 5 073 6 078 7 08 3 8 08 8 9 094 1 099 2 104 3 1094

1 196 124 7 1299 13 5 0 14 01 14 5 2 15 03 15 5 5 1606 1708 1 75 9 18 1 1 18 62 19 13 1964 2016 2067 21 18 2221 2272 23 23 23 74 24 26 24 77 25 28 25 8 0 263 1 2734 278 5 28 3 6 28 8 8 293 9 2990 3 04 2 3 093 3 14 4 3 24 7 3 35 0 3 401 3 4 5 3 3 5 04 3 5 56 3 607 3 65 8

3 76 1 3 8 13 3 8 64 3 9 16 3 967 4 019 4 070 4 1 22 4 173 4 276 4 3 28 4 3 79 44 3 1 4 4 8 2 4 5 34 4 58 5 4 63 7 4 68 8 4 79 1 4 8 94 4 94 6 4 99 8 5 04 9 5 101 5 15 2 5 204 5 3 07 5 3 5 9 5 4 10 54 6 2 5 5 14 5 5 65 5 617 5 669 5 720 5 8 24 5 8 75 5 9 27 5 9 79 603 0 608 2 6 134 6 18 6 623 7

634 1 6 3 92 64 4 4 64 9 6 65 4 8 65 99 665 1 6703 675 5 68 5 8 69 10 6 96 2 7014 7 066 71 17 7 169 7221 7273 73 77 74 28 74 8 0 75 3 2 75 8 4 763 6 768 8 774 0 779 1 , 78 95 794 7 79 99 8 05 1 8 103 8 15 5 8 207 8 25 9 8 3 1 1 8 4 15 8 4 67 8 5 19 8 5 71 8 622 8 674 8 726 8 778 8 8 30

8 9 3 5 8 98 7 9 03 9 909 1 9 14 3 9 19 5 9 24 7 9 29 9 9 35 1 94 5 5 9 5 07 95 5 9 96 11 96 63 97 15 9 768 98 20 98 72

l II“ 9976 *0028 * 008 0 *013 2 *018 5 "o23 7 *028 9 *034 1 03 93 04 98 05 5 0 0602 0654 0706 075 9 08 11 08 6 3 09 15 1 020 1072 1 124 1 17 7 1229 128 1 13 3 3 13 8 6 14 3 8

15 4 3 164 7 1700 175 2 18 04 18 5 7 1909 196 1 2066 21 18 217 1 2223 2276 23 28 23 8 0 24 3 3 24 8 5 264 2 269 5 274 7 28 00 285 2 2905 295 7 3 009 3 1 14 3 219 3 272 3 3 24 3 377 34 29 3 4 8 2 3 5 34 3 64 0 3 69 2 3 74 5 3 79 7 3 8 5 0 3 902 3 9 5 5 4 007 4 060

4 165 4 218 4 270 4 3 23 4 3 76 4 4 28 44 8 1 4 5 3 3 4 5 8 6 4 69 1 4 744 4 79 7 4 84 9 4 902 4 95 5 5 007 5 060 5 1 13 5 218 5 27 1 5 3 24 5 3 76 5 4 29 5 4 8 2 5 5 35 5 5 8 7 5 64 0 5 74 6 5 79 8 5 8 5 1 5 904 5 9 5 7 6009 6 06 2 6 115 6 168 6274 63 26 63 79 64 3 2 64 8 5 65 3 8 65 9 1 664 3 6 696

— A z lo — lo B = 10 l o a — lo 6 lo b+ A = 10 a 6) g a log 6, g b+ g g g , g g (

- N I M V. ADDITION SU BTRACTIO LOGAR TH S .

4 125 4 18 1 4 23 6 4 29 2 4 348 44 04 4 68 3 4 73 8 4 794 4 8 5 0 4 906 4 962 5 24 1 5 297 54 09 5 4 65 5 5 20 5 8 00 5 85 6 5 9 12 5 968 6024 608 0 63 60 64 72 65 28 65 8 4 6640

6920 6976 703 2 708 8 7200 74 8 0 75 3 6 75 93 764 9 7705 776 1 8 04 2 8 098 8 15 4 8 210 8 266 8 3 22 8 603 8 660 8 716 8 77 2 8 8 28 8 88 4 9 166 9 222 9 278 93 3 5 93 9 1 944 7

9 729 9 78 5 984 1 98 98 995 4 *0010 029 2 034 9 04 05 04 6 1 05 18 05 74 08 5 6 09 13 0969 1026 108 2 14 21 14 77 15 3 4 15 90 164 7 198 6 204 3 2099 215 6 2212

25 5 2 2608 2665 2778 28 3 5 3 118 3 175 3 23 1 3 28 8 34 01 3 68 5 3 74 2 3 8 5 5 3 9 12 3 969 4 25 2 4 3 09 4 3 66 4 4 23 4 4 8 0 4 8 20 4 8 77 4 93 4 4 99 1 5 04 8 5 105

5 3 8 9 5 4 4 6 5 5 03 5 5 60 5 6 17 5 674

5 95 8 6015 6072 6 129 6 18 6 ‘ 6 24 3 65 28 65 8 5 664 2 6699 6 75 6 68 13 7098 7 15 5 7212 7269 73 26 73 8 4 7669 77 26 778 3 78 4 0 78 98 795 5

8 24 0 8 298 8 3 5 5 8 4 12 8 5 26 8 8 12 8 8 70 8 9 27 8 98 4 904 1 9099 9 3 8 5 944 2 95 00 96 14 9671 9 95 8 *0015 *0073 *013 0 *018 7 *024 5 05 8 9 064 6 0704 076 1 08 19

1 106 1 163 1221 1278 13 3 6 168 1 173 8 179 6 19 1 1 1968 225 6 23 14 23 71 24 29 24 8 6 25 44 28 3 2 28 90 294 7 3 062 3 120 34 08 3 4 66 3 5 24 3 63 9 3 69 7

3 98 5 4 04 3 4 101 4 15 9 4 274 4 621 4 73 6 4 794 4 8 5 2 5 14 1 5 19 9 5 3 15 5 3 73 5 4 3 1 5 720 5 778 5 8 3 6 5 95 2 6010 629 9 63 5 7 64 15 64 73 65 8 9

68 79 693 7 705 3 7 1 1 1 7 16 9 74 5 9 75 18 763 4 769 2 7 75 0 8 04 0 8 099 8 215 8 27 3 8 3 3 1 8 622 8 68 0 8 73 8 8 797 8 8 5 5 8 9 13 9 204 9 262 93 21 94 95

9 78 7 984 5 99 03 9962 I"OO2O *0078 03 70 04 28 048 7 054 5 0603 0662 095 4 1012 1070 1 129 1 18 7 124 6 15 9 6 165 5 17 13 1772 2123 218 1 224 0 2298 23 5 7

‘ — — — l = — A = lo lo B .. _ l a o 6 A lo a . log a g b, g b+ log og g , log b+ g ( b) V - N I H M 5 1 . ADDITION SU BTRA CTIO LOGAR T S .

2767 28 25 28 84 294 3 3 001 3 3 5 3 3 4 12 34 70 3 5 29 3 5 8 8 394 0 3 998 4 05 7 4 1 16 4 174 4 5 27 4 5 8 5 4 703 4 762 5 1 14 5 173 5 23 2 5 29 1 5 3 5 0

5 703 5 762 5 8 20 5 8 79 5 938 6292 635 1 64 09 64 68 65 27 688 1 694 0 6999 705 8 7 1 17 74 7 1 75 30 75 8 9 7707 8 062 8 121 8 18 0 8 23 9 8 298

8 65 3 8 712 8 77 1 8 8 3 0 8 8 89 9244 93 03 93 63 94 22 948 1 98 3 6 98 96 9 95 5 ‘0014 *0073 0429 04 8 8 054 8 0607 0666 1022 108 2 114 1 1201 1260

16 16 1676 173 5 1795 18 54 221 1 2270 23 3 0 238 9 244 9 28 06 28 65 2925 298 4 3 044 34 01 3 4 6 1 35 20 3 5 8 0 3997 4 05 7 4 117 4 176 4 23 6

45 94 4 654 4 713 4 773 4 8 3 3 5 19 1 5 25 1 5 3 1 1 5 3 70 54 30 5 78 9 5 84 9 5 908 5 968 638 7 65 07 65 67 66 27 698 6 704 6 7106 716 6 7226

75 8 5 764 5 7705 7765 78 25 8 18 5 8 24 5 8 3 05 8 3 65 8 425 8 78 6 8 906 9026 93 8 7 94 4 7 95 07 95 67 9 627 998 8 *004 9 ll"0109 *0169 ‘0229

059 1 065 1 07 11 0771 08 3 2 1 193 125 4 13 14 13 74 14 35 1797 18 5 7 1917 19 78 2038 2400 24 61 25 21 25 8 2 264 2 3 005 3 065 3 126 3 18 6 3 24 7

3 6 10 3 670 3 73 1 3 79 1 3 85 2 4 215 4 276 4 3 3 6 4 3 97 44 5 7 4 8 21 4 8 8 2 4 94 2 5 003 5 064 54 28 5 4 8 8 5 54 9 5 6 10 5 670 603 5 6096 6 15 6 6217 6278

664 2 6703 68 25 68 8 6 725 1 73 12 73 72 74 3 3 74 94 78 5 9 79 20 798 1 8 04 2 8 103 8 469 8 5 3 0 8 5 9 1 8 65 2 8 7 13 9079 9 14 0 9 201 9 26 2

968 9 9 75 0 98 11 98 72 9933 ‘ 03 00 ‘03 6 1 ‘ 04 22 ‘04 8 3 ‘054 5 091 1 09 73 103 4 1 095 1 15 6

15 24 ° 15 8 5 164 6 1707 1768 213 6 2197 225 9 23 20

= l - — — A a lo b l B = lo B = lo lo lo A = 10 a b . og g , og b+ g g a g 6, g 0+ g ( ) . 52 V - N H M . ADDITION SUBTRACTIO LOGARIT S.

97 17 9 78 4 98 5 0 99 16 998 2 l“004 8 *0113 03 77 044 3 05 08 05 74 064 0 0706 1034 1099 1 165 123 0 1296 13 6 1 14 27 168 8 175 3 18 19 18 8 4 1949 2014 208 0 23 40 24 05 24 70 25 3 5 2600 2665 273 0

2990 3 05 4 3 119 3 18 4 3 3 13 3 3 78

‘ 3 63 6 3 701 3 766 3 8 95 3 95 9 4 024 4 28 1 43 4 5 4 4 74 4 5 3 8 4 6 02 4 667 4 923 498 7 5 05 1 5 115 5 179 5 24 3 5 307 5 5 63 5 627 5 69 0 5 75 4 5 8 18 5 8 8 2 5 945

6200 6 264 63 27 63 9 1 64 5 4 65 18 65 8 1 68 3 5 68 98 6 962 7025 708 8 7 15 1 7215 75 3 1 75 94 765 7 7720 778 3 784 6 8 098 8 160 8 28 6 8 3 49 84 12 8 4 74 8 7 25 8 78 8 8 8 5 1 8 9 13 8 976 903 8 9 101

9 3 5 1 94 75 95 3 8 9600 966 2 9725 99 74 *003 6 *0098 *0160 *0222 *0284 *03 4 6 05 94 065 6 07 18 08 4 2 09 04 0966 1213 1275 13 3 6 13 98 14 60 15 21 15 8 3 18 29 18 9 1 195 2 2075 213 6 2198

24 4 3 25 04 25 65 268 8 274 9 28 10 305 4 3 116 3 177 3 298 3 3 5 9 3 4 20 3 725 3 78 5 38 4 6 3 907 3 968 4 028 4 271 4 3 3 1 4 39 2 44 5 3 4 5 13 4 5 74 4 8 76 4 996 5 057 5 1 17 5 177 5 23 8

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5 13 - 167 7 - 3 11 425 575 8550 3 - 11 427 545 5712 3 - 19 - 4 7

3 - 727 3 7 - 5 9 7 - 3 8 3 425 5205727 420257 5554

’ 7 - 3 13 3 - 17 - 4 3 3 - 13 - 23 45025 5 5554 3 - 29 - 3 1 45 1 2025545

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23 - 97 849 665 9841 455 521 7001 7 - 1 7 - 23 3 - 1 1 - 8 3

3 3 - 8 3 850829 2786 3 - 7 - 107 13 - 1 73 4 1 450174 7505

9 3 - 75 1 3 7 - 6 1 3 - 25 1 3 - 7 - 13 1 480805 2114 3 - 9 19 3 1 - 8 9

’ 7 - 17 - 19 3 1 - 73 855 45 1 5901 855 11 - 25 1 3 - 3 07 4420001501 3 - 13 - 7 1

, 3 - 75 7 3 - 1 1 - 23 4 3 - 5 3 17 - 16 3 4 7 - 5 9 448 575 8708 7 - 3 9 7

’ ’ 558 125 2855 3 - 76 1 859 266 1646 3 - 7 - 109 3 - 103 11 - 23 3 - 9 29 445 445 5 148

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055 7000455 7 - 3 3 1 3 - 773 20- 07 45 0005 0750

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7 3 - 7 - 3 7 857 014 7055 3 - 19 - 4 1 869 0802218 19 - 14 9 452245 5745 17 - 167 “ 550401 4187 870518 0896 3 - 29 3 - 94 7 458 775 8507 3 - 13 - 73 7 - 1 1 - 3 7

’ 13 - 18 1 5725505525 7 - 33 7 454 007 2178 3 - 3 17 455 0102104

’ 3 - 78 7 17 - 13 9 3 - 263 23 - 103 455 5 17 8575 7 4 09 19 - 15 1

9 9 3 - 7 1 13 875 0201817 3 43 6 1 3 - 1 1 - 29 13 - 17 3 - 7 - 13 7 45 024 1 55 10 Q 876 769 896! 877 124 0428 078 215 1407 4 3 - 67 5 . 8 1 450445 7550

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052 107 2104 19 - 127 885 276 6504 4 1 - 5 9 3 071 454 055 4001 - 13 9

9 3 269 054 555 4141 3 - 8 09 7 - 3 4 7 3 7 - 79 455 422 7224 20- 101

1 1 - 13 - 17 3 - 8 1 1 855 855 5202 3 - 9 77 7 - 4 19 3 - 1 1 - 8 9 458 100585 1

3 5 57 557 7704 3 1 - 79 17 - 173 3 - 109 7 4 21 3 - 98 3

3 4 3 3 - 7 - 13 500755 5287 13 - 227 470255 4470 470851 8045 1 1 - 269

' 23 107 502 1501405 3 - 8 23 3 - 7 - 4 7 471 75 1 5515 472 5101075 ’ 3 7 - 6 7 472002 55 15 3 - 001 - 3 - 3 3 1

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‘ 4 7 5 3 1 1 - 227 - l 7 4 1 7 8 3 - 3 7 475 075 4555

z 1 29 9 9 5 5 00 4 3 . 2 3 0 0 7 . 07 0 log . log

122 I X P I M AND MP E M E R NU . . E CO OSIT B RS

3 502105 55 14 602886 6901 602819 8424 19 - 211 7 - 64 3 3 49 49 558 887 558 1 3 - 167

3 - 7 - 19 1 608 469 1607 3 . 13 - 103 604 118 0062 13 - 3 4 7 554 455 8885 554 555 04254 18

3 504 554 078 1 3 - 14 9 604 98 1 6296 - 79 3 - 1 1 - 13 7 555 425 5877 7 - 647

3 7 - 109 1 1 - 3 6 7 7 - 5 77 23 - 19 7 3 - 15 1 1 13 - 349 3 - 17 - 8 9 ’ 3 - 44 9 13 - 3 1 1 3 - 19 - 7 1 6477847 7768 7 - 11 - 5 9 557 724 9542 557 915 9858

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3 3 - 23 - 5 9 500014 4101 3 . 15 1 010668 7053 7 - 663 17 - 269 23 - 19 9 19 - 24 1

9 7 - 1 1 - 5 3 3 - 13 6 1 6 1 - 6 7 3 - 29 - 4 7 8 - 5 09 551 149 8572 3 - 1 1 - 13 9

611 829 4796 512041 744 5 17 24 1 5 12077 9155 55 1 907 2928 552474 5088 3 ’

’ 3 - 13 67 1 1 3 73 3 - 3 7 7 - 5 8 7 4 3 - 107 558 0409749 17 - 271 1 1 - 4 19

5 618 947 4768 23 - 179 3 - 13 73 - 5 3 7 - 65 9 3 3 1 - 14 9

13 3 17 7 - 19 - 3 1 616 684 4689 616 844 8829 554 785 9585 3 - 23 - 67 7 - 66 1 3 - 15 - 4 3

‘ 33 - 17 616 266 4068 3 - 7 197 616 896 4264 1 1 - 4 21 4 1 - 1 13 555 28 7 0950 555 424 8 725

’ 4 1 - 101 3 - 13 8 1 1 1 - 13 - 29 3 - 4 6 1 3 - 7 - 13 - 17 555 798 5887 3 - 154 9

’ 7 - 5 93 618 861 9811 618 7800246 618 988 9204 557 545 8595 3 - 1 1 - 4 7 558 105 2579

9 3 - 19 - 73 23 - 18 1 3 - 4 63 1 1 - 3 79 59 - 79 558 555 4155 13 - 3 5 9 7 - 23 - 29

4 3 - 97 3 - 13 - 107 690864 4768 3 - 7 - 199 509 595 7810 570155 0452

3 7 - 113 4 7 - 8 9 5 3 - 79 3 - 7 - 223

3 - 11 - 127 7 - 5 9 9 3 - 13 99 13 - 17 - 19 571 255 4029

’ 525 252 5515 3 o 4 67 7 - 601 3 - 23 o 6 1 3 - 15 67 579 874 9787 17 - 277 9 524 555 2414 1 1 - 3 8 3 626 008 6010 7 - 673 3 - 15 7 1 5 3 - 8 9 3 4 1 1 - 13

9 3 7 67 4 1 103 3 14 09 626 28 7 6861 574 217 9455 29 - 1 63 574 7598 140 3 025 448 0255 3 - 17 - 8 3 19 - 223 3 - 15 7 3 - 19 - 8 3 575 185 5045 7 4 77 ’ 527 455 2725 627 678 0818 3 1 - 13 7 7 - 607 11 - 4 3 1 3 o l 7 - 3 1 4 7 - 101 3 - 158 3

’ ’ 3 - 13 - 109 525 505 5527 3 - 1 1 - 4 3 520507 5401 7 - 97 5 7 - 71 577 5 15 7048 9 629 611 6842 3 - 7 - 29 17 - 25 1 3 - 14 23 19 - 25 1 4 7 - 13 - 4 7 1 1 - 3 8 9 13 - 3 67 3 - 3 7 - 4 3 17 - 28 1 3 - 5 9

3 - 14 27 581 748 0744 3 - 14 29 7 - 5 8 3 579 7005809 580244 8870

' 7 - 6 13 555 155 5557 3 - 14 33 580507 4290 3 - 13 - 4 1 581 1507499

1 1 - 23 69 - 73 3 1 - 139 581 88 1 7050 3 - 1601 11 - 19 - 23 3 - 7 - 229 ’ 3 - 4 79 19 - 227 3 - 14 3 9 7 - 6 17 17 - 28 3 582415 85 17 582775 5458 5 1 - 79

’ 29 o l 4 9 3 - 1 1 - 13 1 586 186 8052 3 o l 3 - 3 7 3 - 15 07 3 - 1609 1 1 4 39

3 6 1 - 7 1 7 - 6 19 557 1504221 58 7 559 5501 584 087 08 75 3 - 179 7 - 69 1

3 3 - 144 7 4 3 - 101 3 - 7 - 23 4 7 - 103 29 - 167 3 7 - 13 1 13 - 3 73

19 - 229 3 - 14 5 1 689 187 6699 3 14 63 23 - 21 1 3 - 1 6 19 4 3 - 118

9 ’ 7 - 8 9 689 785 2180 1 1 - 3 9 7 17 - 25 7 585 725 5211 3 - 15 21 3 - 5 4 1

3 - 3 1 - 4 7 640779 4778 3 14 5 9 29 15 1 587 518 1295 11 - 44 3 588 1527555 7 - 17 - 4 1

9 3 13 - 33 7 3 - 4 8 7 4 1 - 107 B - 7 - 11 - 19 3 - 1627 19 - 25 7 3 - 18 1 “2200878

642 668 4871 23 - 19 1 648 166 4666 5 3 - 8 3 67 - 73 15 4 - 23 3 5 9 - 8 3 3 - 23 - 7 1

’ 9 3 - 163 7 - 17 - 3 7 3 - 13 - 1 13 13 - 29 500451 8002 7 - 701

’ 1 1 4 01 3 - 14 7 1 7 - 63 1 3 - 4 9 1 3 - 163 7 3 4 1 44 9 591 875 8225 ’ 646 6206149 646 716 9894 19 - 23 3 43 - 103 7 - 19 - 3 7 3 - 5 4 7 13 - 3 79 3 - 3 1 - 5 3

3 - 7 - 21 1 23 - 19 3 592085 0025 505 111 1155 1 1 4 49

‘ * 647 4807782 3 c 148 1 3 - 14 8 3 3 - 6 1 598 9905105 - 9 7 7 - 101

3 648 467 6948 6 1 - 73 5400425541 7 - 13 ao4 a 5 13 427 505 218 9180 ’ 3 148 7 5405205555 4 1 - 109 l l - 4 l 7 - 709 505

9 ° 9 17 o 263 3 - 7 - 7 1 1 1 - 3 7 3 505 518 4592 3 - 7 - 79

661 874 9489 661 668 7889 7 - 64 1 3 41 45 1 597 8808582 3 - 1663 ’ 3 - 4 99 662 68 6 4186 3 - 14 99 1 1 - 4 09 7 - 23 - 3 1 698 861 6661 19 263

029 995 7 5 97000 . 43 log log . 5‘

D I T M E 12t. X P I ME AN COM PO U . I . R S E N B RS

w9 2805029 3 - 168 9 699 751 08 17 740441 6450 740599 5128 740915 0765 7 - 78 7

’ ’ 3 - 5 5 7 29 - 173 3 - 7 - 23 9 3 11 - 167 3 - 6 13 741 8508041

7007902214 700968 1782 1 1 - 4 5 7 4 7 - 107 742017 7471 3 7 25 3 7424804545

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' 3 ’ 3 - 4 1 3 - 1 1 - 17 3 - 18 4 7 23 - 24 1 3 - 4 3 3 1 - 179

708 877 8685 3 1 - 163 13 - 3 8 9 704 064 6764 7 13 - 6 1 744 8403908 3 0 17 - 109 1

’ 3 - 7 - 24 1 6 1 - 8 3 3 - 5 63 3 7 - 13 7 5 7 - 8 3 745 809 0599 19 293 745 777 2179

1 ’ 11 - 4 6 1 3 - 19 - 8 9 705 607 1684 3 - 1693 3 - 5 19 745 089 0481 3 - 11 - 13 7 - 79 7

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9 708 165 8579 3 13 - 13 1 3 - 18 67 13 - 4 3 1 3 - 7 - 8 9 71 - 79

19 269 708 675 7927 7 - 17 - 4 3 709 185 1295 3 1 - 18 1 3 - 18 7 1 4 1 - 13 7 3 48 73

9 3 - 5 6 9 4 7 - 109 23 - 223 7 - 1 1 - 73 740958 0855 1 7 - 3 3 1 13 - 4 3 3

' 7 - 73 3 3 - 29 - 5 9 11 - 4 67 3 - 5 71 3 - 18 77 4 3 - 13 1 3 - 18 79 751 2020945

5 3 - 9 7 3 7 - 13 9 711 554 1588 19 - 271 una8 5 mn7 75 1 817 7877 3 - 7 - 269

3 3 - 17 - 101 7120601425 3 - 19 1 7 - 1 1 - 67 752 125 8078 752 278 9855 752585 1787 752 789 5989 ’ 13 - 3 9 7 3 - 1721 718 288 4615 3 1723 3 o l 7 - 3 7 7 . 8 09 3 18 8 9 758 505 45 70

9 718 574 5878 7 73 9 3 1 - 16 7 714 245 9110 3 - 3 1 - 6 1 7 - 8 11 3 - 63 1 * 3 - 1 1 - 15 7 7 1 - 73 3 - 7 715 088 6707 13 - 19 - 23 754 577 5550 1 1 - 4 7 755 055 0888 ’ 29 - 179 3 o 5 77 715 7527168 3 173 3 3 - 7 - 27 1 755 841 1888 4 1 - 13 9

9 3 - - - 7 14 3 0 1 1 4 3 4 1 127 715 754 8574 755 95 1 0410 3 1901 13 4 3 9 3 - 1 1 - 173

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’ ’ 3 - 174 7 7 - 107 3 - 1 1 - 5 3 29 - 18 1 758 987 5469 750188 8158 7 - 8 21 759 5928086 !

‘ 5 9 - 8 9 3 - 17 - 103 7 - 75 1 3 - 1763 3 o 7 l 1 1 5 23 3 - 19 - 101 13 - 4 4 3 1 721 058 8018 19 277 23 - 229 1 1 - 4 79 7 - 8 23 73 - 79 3 - 64 1

3 - 7 - 25 1 722 057 7718 3 - 175 9 722551 6620 29 - 199 23 25 1 5 3 109 751 8526945

’ 722 715 1575 3 4 1 4 3 3 - 4 1 - 4 7 752 158 1928 3 - 64 3 7 - 8 27

724 029 9729 7 - 75 7 762758 5649 3 - 193 1 1 1 - 17 - 3 1 3 - 193 3

9 3 . 19 3 1 724 521 5271 3 - 29 - 6 1 725 0127258 768 502 8655 7 - 8 29 755 951 8250

3 ’ 4 7 - 1 13 3 0 7 - 1 1 - 23 13 - 4 09 3 - 197 3 - 13 - 14 9 75 4 4008280 3 - 7 - 277 1 1 - 23 9 17 - 3 13 725 155 4052 764 997 5998 3 5 47 755 445 0181 3 - 29 - 5 7

9 3 3 - 1777 726 971 583 7 3 - 5 93 19 - 28 1 7 - 17 19 - 3 07 13 4 4 9 755 888 4758 1

9 3 - 13 - 13 7 728 1101841 3 478 3 3 11 - 5 9 755 585 8858 3 - 194 9 757 08 1 5214

3 728 484 9510 1 1 - 4 8 7 23 - 23 3 767 2800981 3 - 195 1 757 575 2240 3 - 7 o 3 1

3 - 178 7 3 1 - 173 3 - 1789 7 - 13 - 5 9 767 971 7214 1 1 - 13 - 4 1 758 415 0882 758 554 1095

3 * 4 1 - 13 1 3 - 199 19 - 28 3 3 - 1 1 - 163 3 19 103 7 - 8 3 9 3 - 6 5 3 759 808 4502

’ 7808529020 78 1 846 9755 17 - 3 17 769 45 1 1794 3 - 3 7 - 5 3 7 - 29 3 - 13 - 16 1

9 3 - 5 99 78 1 8804208 3 - 7 . 25 7 7828 18 8275 4 3 13 7 71 - 8 3 77068 1 1278

9 1 1 4 9 1 3 - 18 01 782055 8595 3 - 5 01 3 - 7 - 28 1 771 0727882 3 41 479 19 3 1 1

4 7 - 773 788 488 0271 788 758 8855 788 9101510 23 - 25 7 3 - 73 6 1 - 97

‘ 3 - 13 - 13 9 1 1 - 17 - 29 3 - 6 7 6 1 - 8 9 3 1 - 19 1 772541 7825 772884 9272

’ 3 - 18 1 1 785 859 8880 3 o 65 9 17 - 3 49 3 - 19 79 778 718 8258

’ 785 678 7256 785 888 88 58 13 - 4 19 785 5 15 8079 13 - 4 5 7 3 - 7 - 28 3 19 - 3 13 3 - 66 1

3 - 23 - 79 7 - 19 - 4 1 3 - 17 - 107 5 3 - 103 11 - 5 4 1 774 785 8825 7 - 23 - 3 7 5 9 - 101

9 3 4 3 - 127 3 - 607 7 - 11 - 7 1 3 - 18 23 3 - 198 7 67 - 8 9 3 - 13 - 17 4 7 - 127

788 055 7148 13 - 4 21 788 5-127409 788 701 8004 7 - 8 5 3 - 18 1 4 3 - 13 9 3 - 1993

8 3 - 7 - 29 789 018 2459 3 - 3 1 - 5 9 11 - 4 99 776 778 8024 3 1 193 777 209 25 81 5 3 - 1 13

9 17 - 19 3 - 18 3 1 23 - 23 9 3 - 199 7 13 4 6 1 3 1999 7 o 8 5 7

029 995 7. 5 9 70004 log log 3 . 124 I P I M M P X . R AN D E N E E CO OSIT UM B RS.

3 7 0 i

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19 - 3 17 780245 2889 814 8 14 2002 1 1 - 5 93 6 1 - 107 8 14 845 5585

' 3 7 - 163 3 - 2011 780821 1759 3 o 1 1 - 6 1 - 3 11 4 7 - 13 9 13 - 5 03

7 - 8 63 781 252 5942 781 589 9686 23 - 263 3 1 - 211 8 15 0428409 3 - 3 7 - 5 9

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13 4 67 788 408 2811 5 9 - 103 817 681 4672 3 - 7 - 3 13 818 027 8419

- 3 2027 7 - 1 1 - 79 3 - 2029 784 545 9741 818 291 8908 29 0 227 7 - 94 1 11 - 5 99

9 5 ’ 784 688 5995 3 - 677 7 - 13 - 67 3 19 107 3 13 19 34 7 3 73 3

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788 945 7270 3 - 7 - 29 3 4 7 - 13 1 3 - 205 3 825 017 5204 3 828 409 0149

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3 - 208 7 796 782 4117 3 - 208 9 797 198 2698 8800109859 880189 8874 5 7 - 101 7 - 9 5 7 5 9 ' 797 886 8008 3 - 17 - 4 1 797 7521287 3 - 7 - 13 - 23 3 - 3 7 - 6 1 13 - 5 21 33 - 25 1 88 1 155 5889 !

1 1 - 5 71 6 1 - 103 798 448 4604 19 - 3 3 1 88 1 298 7444 3 - 7 - 17 - 19 1 1 - 6 17 3 - 3 1 - 73

7 3 - 2099 799 271 6088 852051 5145 13 - 5 23

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3 - 29 - 73 808 978 8558 3 - 13 - 163 808 888 8250 7 - 1 1 - 8 9 005 104 1495

7 7 3 - 7 - 101 808 984 8499 3 - 228 7 88 6 5 18 9989 3 - 7 - 109 885 895 5 154

23 277 804 848 9185 7 - 9 11 804 7526022 007 019 0105

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9 3 - 213 7 7 - 13 1 839 5408980 3 1 - 223 889 917 7757 - 3 7

9 9 807 602 6699 3 214 1 808 008 2999 3 - 214 3 3 - 769 7 - 23 - 4 3 3 - 2309 13 - 4 1 ' 5 9 - 109 7 - 9 19 3 - 23 11 3 o 26 7

3 - 19 - 1 13 3 - 7 - 3 07 809 4028700 11 - 63 1 5 3 - 13 1 84 1 797 2989 84 1 0028 117

3 809 627 04 19 3 23 9 1 1 o 5 8 7 3 - 215 3 3 - 7 - 3 3 1 17 - 4 09 8425 46 8865

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9 811 5420215 13 - 4 99 3 - 7 - 103 3 - 13 - 179 844 0420420 3 - 17 - 13 7 29 - 24 1

3 8128 11 5091 4 3 - 15 1 73 - 8 9 6 7 - 97 844 589 8021 3 - 7 - 3 7 844 911 8789 3 - 23 3 3

2 9 5 00043 0 9 95 97 . log 70 log

126 I X P I M AN D MP E M E . R NU E CO OSIT B RS.

- 127 3 - 17 - 15 7 900578 2985 929 4700152 5 7 - 127

908 585 78 17 904 011 8885 3 - 28 3 7 3 - 17 - 167 7 - 1217

9 13 017 71 1 13 7 - 3 1 - 3 7 3 - 94 7 980795 2580 3 - 28 4 3

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3 8 3 - 97 905 057 0991 905 281 1558 17 - 603 3 - 28 5 1 4 3 - 199 3 - 3 17

3 268 7 11 - 73 3 3 28 8 9 905 0107155 7 - 1223 13 - 65 9 11 - 19 - 4 1

3 3 - 13 - 23 4 1 - 197 3 - 2693 3 . 28 5 7 23 0 3 73

007 455 1058 907 787 4481 907 804 8854 19 3 - 28 6 1 3 1 - 277 3 - 7 - 4 09

9 ’ 3 - 29 - 3 1 908 109 5404 3 - 2699 7 - 13 - 8 9 1 1 o 7l 13 - 66 1 984 447 “

’ ’ 908 5805822 3 - 3 7 - 78 11 - 67 3 - 17 - 5 3 3 - 4 7 - 6 1 7 - 1229 3 - 19 - 15 1 994 969 7078

3 9 909 074 4014 7 - 19 - 6 1 909 895 5450 23 - 3 5 3 79 - 109 3 - 11 - 29 7 - 123 1 3 - 13 - 17

909 715 4582 - 4 3 3 7 - 23 3

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914 924 0489 19 - 4 3 3 1 1 43 - 131 3 - 2909 - 4 3

915 4525017 915 558 1154 915 7500550 - 107 941 058 9882 3 - 4 1 - 7 1

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917 187 7528 7 - 1 18 1 917 4529919 3 - 23 - 127 1 1 - 79 7 3 - 3 7 - 79

917 558 0248 3 17 - 4 8 7 3 1 o 28 3 67 13 1

’ ’ ‘ 7 - 13 3 - 11 - 25 1 918 897 5888 3 - 3 07 3 - 2927 948 049 8898 3 - 29 - 101 1 1 - 1 7 - 4 7

918 505 9151 918 711 5554 918 921 0901 4 3 - 193 5 9 - 14 9 19 - 4 63 3 - 7 - 4 19

’ ’ 3 - 2767 L9 - 23 3 23 - 3 8 3 3 919 558 2825 3 - 17 - 18 3 019 005 7015 3 - 4 7 - 5 9 3 - 1 1 - 8 9 7 - 1 269 3 - 293 9 945 419 8425

‘ 5 3 - 15 7 - 4 1 11 - 767 945 517 8291 3 - 17 - 1 73 7 - l 3 o 9 7 33 - 1 09

’ - 3 97 3 1 - 269 11 - 73 945 945 408 1889

’ 9 19 - 4 3 9 17 - 4 9 1 3 - 11 - 23 - 4 21 3 7 - 23 9 3 - 98 3 945 094 1951

7 - 1193 6 1 - 13 7 5 3 - 167 3 - 13 - 227 17 - 5 21 3 - 29 63

9 9 3 - 9 29 3 - 278 9 922 947 7707086 7 - 18 1

9 11 - 76 1 928 0805154 3 3 - 296 7 19 - 4 67 3 - 11 - 269 13 - 68 3

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’ 3 - 7 - 401 925 4008007 3 - 5 3 995 776 0588 1 1 - 8 11 9606108980 79 - 1 13

9 925 879 0898 3 - 93 7 11 - 13 - 5 9 - 9 7 3 - 13 - 229 7 - 127 7

23 - 3 67 990496 7898 926 7024942 7 0 17 - 7 1 3 - 11 - 271 3 - 19 - 15 7

3 * 3 - 3 13 79 - 107 3 - 28 19 11 - 75 9 95 1 871 5571 7 - 1279 17 - 3 1

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5 9 7 - 1213 3 - 19 - 14 9 29 - 293 3 - 28 3 3 3 - 3 7 17 23 3 2999 954 194 25 10

029 995 log 7. I X P M E AND MP E N MB E . RI CO OSIT U RS. 127

964 2907617 3 - 3 001 - 11 - l 3 3 - 3 16 7 123 3 - 3 169 3 7 - 25 7

95 4 779 9897 954 059 0711 978 225 1817 3

3 - 3 1 - 97 7 - 128 9 955 009 0500 978 5825559 8 9 - 107 7 - 13 6 1 13 - 73 3

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9 9 5 5 . 0700 4 3 0 . log 02 9 7. log 128 I X P I M E AND M P I T N E . CO O M R S E U B RS.

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9 124 0527204 5 7 0 19 9 125 058 1512 125 128 2726 140058 5818 3 0 29 0 5 3 101 0 13 7 3 0 7 0 65 9

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0 3 0 4 4 5 1 3 0 6 1 0 73 3 0 19 7 0 1979 3 0 3 1 - 14 9 141 78 1 8948

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103 0 13 1 3 0 11 0 4 09 180801 5978 17 0 8 23 7 0 1999 140084 9626 146 097 0185

lo 0004 g 07 3 . 132 I X PR I M E AND M P E N E . R CO OSIT UMB S.

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03 147 1505252 3 0 4 679 101 - 139 11 0 13 21 152855 2785 152474 7904 7 0 3 1 0 5 7 I 9 0 0 I 04 19 0 73 9 3 0 3 1 0 15 1 1 1 1277 3 0 7 0 223 3 3 7 0 13 1 152554 0041 3 0 13 0 3 73 159 8521480

05 147 707 2888 13 0 23 0 4 7 17 0 8 27 159 892 0407 155 071 0090 3 0 23 0 211 3 06 3 0 4 3 0 109 7 0 4 1 1 1 0 1279 158 191 9019 159 2508495 7 0 208 1 17 0 85 7 ’ 07 148 824 9680 3 0 4 69 1 7 0 201 1 3 0 13 0 19 13 0 19 0 5 9 3 0 4 3 0 113 6 1 0 23 9

08 148 $8 0 85 148 695 1798 148 818 5146 73 0 193 7 0 208 3 3 0 4 8 5 1 29 0 5 03

09 3 0 7 0 11 0 6 1 17 0 8 29 3 0 3 7 0 127 23 0 6 13 154 144 5025 1 1 0 13 117 13 0 11 23

9 0 10 5 9 0 23 9 3 0 15 67 149 454 5508 3 0 4 703 3 0 3 1 0 15 7 17 0 8 69 3 0 5 4 1 7 0 2097

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3 0 1 2 3 0 5 23 29 0 4 8 7 3 0 17 0 277 7 1 0 199 151 977 0771 7 0 208 9 155 155 9514 155 914 5590

13 13 0 108 7 3 0 7 0 6 73 6 7 0 21 1 3 0 4 9 77 155 9009725 3 0 7 0 17 0 4 1 165 611 4108 i 2 14 79 0 179 150541 5414 7 0 4 3 0 4 7 150725 7455 3 0 15 27 97 0 15 1 3 0 19 0 25 7

7 15 3 0 5 3 0 8 9 1508-18 6067 15 1 0825010 7 0 13 0 23 155 925 5495 155 045 0879 107 0 137

4 16 3 0 4 721 3 1 0 4 5 7 3 0 4 723 3 0 18 1 11 0 3 1 0 4 3 3 0 48 8 9 155 4005055

17 3 7 0 38 3 151 451 7871 1 1 0 128 9 17 0 8 63 3 0 67 0 73 13 0 1129

18 3 0 29 0 163 13 0 109 1 3 0 4 729 7 0 2027 5 3 0 277 155 914 7980 19 0 773 3 7 0 3 97

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