' d ce 60 ontl EMh Bo s. lomo . ar Pri C .

Wh am whe n th e Sub ec t Dem and s . Amp ly j

N6 k— M U - . CHI NEYS FOR F RNACES AND STEAM BOIL

E RS. . r s r C E . 3d e ed By R A m t ong , . Am rican . Re vis ed art l w w A n dix and y re ritten , ith an p T ghim ne E M. on heory of y Draught , by F . E . Ide . — — No . 2. S X Ze m h TEAM BOILER E PLOSIONS . By Colburn . Ne w s r . s edition , revi ed by P of R . H . Thur ton . — - 3 . O F I Wo . PRACTICAL DESIGNING RETAIN NG WALLS.

ed ltion . W i Fourth . By Prof . Ca n . " — o 4. F N . PROPO RTIONS O PINS USED IN BRIDGES By

s . E . e e C . E . 2d e t w x Cha B nd r , di ion , ith appendi .

. F. utler. BUILDINGS By W . B c ec ition e ~e ite and e l r b ames Se ond , r d d n a ged y J

e . L . Greenl af. C . E — A D 6. N N No . O THE DESIGNING CONSTRUC TION OF b . B STORAGE RESERVOIRS yArthur Jaco , A.

. ec o e Wi e t t n b E . B S nd edition , r vis d , h addi io s y Sherman Gould . — AND E 7. N NO. A RE SURCH RGE-D DIFF RE T FORMS OF v TAIN G . m e . C . E . IN WALLS By Ja s S Tate , - No . 8. A N TREATISE ON THE COMPOU D ENGINE . By

u un . e e n e e John T rnbull , j S cond ditio , r vis d by ns Prof . S . W . Robi on . - NO . 9. A . th b tt TREATISE ON FUEL By Ar ur V. A bo ,

C . E . u the lt e t ll Fo nded on origina r a ise of C . Wi

m s . ia . Siemen , D C L . — 0 0. N N . No . 1 COMPOUND E GI ES Trans lated from the

e c . e . e i e s Fr n h of A Mall t Second d tion , r vi ed , w r t r c ct ith esul s of Ame i an Pra ice by Ric hard H.

C . E . Buel , - T o r . w No. Ila . . All n HEORY ARCHES By Prof a . - 2. U . No . 1 THEORY OF VO SSOIR ARCHES Prof. W . ‘ 2 . r se ar ed . Cain Second edition , evi d and en g — MET IN - m G . J No . ASES WITH COAL MINES By . J.

k s . r se e l At in on Thi d edition , revi d and n arge d

b w B . m s un y Ed ard Willia , j . ‘ RICTION . J. J . h H OF AIR IN MINES By Atkinson . e t Seco nd Am eric an di ion .

. Illustr M KE W E . . . . e C . E . No . ARCH S By Prof E W Hyd , 1 6 — A P E O FOR O No . . GRA HIC M TH D S LVING CERTAIN L Q UESTIO NS IN ARITHMETIC O R A GEBRA .

s . By Prof . G . L . Vo e — L . . WATE R S PP Y . WATER AND U . By Prof W H

o rfle ld U s on. C of the niver ity College , Lond Sec ond American edition . — B m W AND E U O . y SE ERAGE SEWAG P RIFICATI N ” k r ss o r i eri ew . M. N. Ba e , A ociate Edit Eng ne ng N s 9 - M Ne. 1 . RSE STRENGTH OF BEA S UNDER TRANS“VE

AD . l t . eor LO S By Prof W. A lan , au hor of T y ” h

es . c d r s . of Arch Se ond e ition , evi ed — ‘ No . 0. A B . 2 BRIDGE ND TUNNEL CE NTRES . By John Mc Mas te r ti . C . E . c e , Se ond di on - A si . L econ t on. ichard up. S FETY VA VES . S d Edi i By R E H. . C . Buel , ’ — M SO Y E r ul zz. I R . . , Ito. H GH A N DAMS By She man Go d M E . . C . Am Soc . . — 23 . TH No . E FATIGUE OF METALS UNDER REPEATED r STRAINS . With va ious Tables of Re s ults and x im e s om the m w . e E per nt Fr G r an of Prof . Lud ig n en bur h w r g , c . . r e i g ith a P efa e by S H Sh ev , — No . 2 A 4. PRACTICAL TREATISE ON THE TEETH OF

. . . ns . eco n WHEELS By Prof. S W Robi on S d e n se ditio , revi d .

— HE HO Y AN D L U O O F C ON No 25 O N T . . T E CA C LATI R N D M l x Ph . . B R . . T IN UOUS BRIDGES . y Wi co , zit—PRACTICAL TREATISE ON THE PROPER TIES ' g c o lNUOUS BRIDGES. By Char les g d gfi n er , - A N. No . 27 N C N ND . O BOILER IN RUSTATIO CORROSIO w w B F. J . . N e . R v F E e . b . . 1 08 1 y Ro an Ed y 1. — E 28 . AN R O S. No . TR SMISSION OF POW R BY WI E R PE

ec . e r . . . . S ond edition By Alb t W Stahl , U S N — R m the e 29. M E . n l te nch No . STEA INJ CTO S Tra s a d fro Fr of M . Leon Pochet . — AL ‘ MA NE TISM THE MA No . 30. TE RREST RI G AND G I M O F Prof NET S IRON VESSELS. By . Fair m an Rogers . — 31 . R N F N No . THE SANITA Y CONDITIO O DWELLI G - HOUSES IN TOWN AND COUNTRY . By

e r . un . G o ge E Waring , j — - No 32. M K . CABLE A ING FOR SUSPENSION BRIDGES.

. C . E . By W Hildebrand , —M No. 33. e . ECHANICS OF VENTILATION . By Geo rg W r a C . E . Ne w e i e i on R fte , and R v s d Ed ti . - No . a ard . E . . ud C . s G FOUNDATIONS By Prof Jule ,

r t h c . Second edition . T ans lated rom t e Fren h — e 85. AN E ID N . THE RO BAROMETER : ITS CONSTRUWO . e e e TION AND USE Com pil d by G org _ Plym pton . Eighth edition . - No . Si M R le k Mxwell m AND J. ATTE MOTION . By C r a ,

. . r e M A Second Ame ican dition . — No . 37. G U S EOGRAPHICAL S RVEYING ITS USE , k De METHODS , AND RESULTS . By Fran Yeaux r C . E . Carpente , — M 5 0 . 88. X M D RI GE MA I UM STRESSES IN FRA E B D S.

m A. B Prof. ll H. . E . y Wi ia Cain , , C New re i s and v ed edition. N -A K - G O. fl HANDBOO OF THE ELECTRO MA NETIO r TELEGRAPH . By A . E . Lo ing .

No fl — O SS . TRANSMISSION OF POWER BY C MPRE ED

. r Z r . . e e t n AIR By Robe t ahne , M E S cond di io . — N lL S . l m Ke t O. TRENGTH OF MATERIALS By Wi lia n ,

C. E . s s . E i r E n i ne er i n ews . n E N d . , A oc d to , g g Seco d 42 — O Y OF - O OF No . . THE R STEEL C NCRETE ARCHES , AND

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. . e Columbia College Second edition R vised. — 45 . B . . . No . . THERMODYNAMICS y Prof H T Eddy. Uni vers ity of Cinc innati . - - 46 K . r m the e ch of No. . ICE MA ING MACHINES F o Fr n

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i C . E . By William Ca n , — 49 ON UI . No . . THE MOTION OF A SOLID IN A FL D r D s Ph. By Thoma C aig, . 50 — - : N C ON No . . DWELLING HOUSES THEIR SMA ITARY U E N . STR CTION AND ARRANGE TS. By Prof ld . C o rfi e . W. H — 5l. : N ETC . NO. THE TELESCOPE ITS CONSTRUCTIO , By Thomas Nolan . — 58 . sl e m the No . . IMAGINARY Q UANTITIES Tran at d fro r r French of M . A gand . By Prof. Ha dy . — M 58. N : NO . I DUCTION COILS HOW ADE AND HOW USED . Fifth edition . — NO fi K N . . Ken . I EMATICS OF MACHINERY By Prof n r c b edy. With an int odu tion y Prof. Thurs to n . — D 55 E AN N. No . . SEWER GASES THEIR NATUR ORIGI

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P E F AC E R .

THE following pages were prepared as ’ o f a part the author s lectures on Geodesy , given to succeeding classes in the Uni

o f li versity Il nois , and are now published for the greater conveni ence o f his stu the e dents , and with hope that th y may The be useful to others . author does not claim that there is anything new o r original in this volume ; he has simply condensed into a single book what here tofore could be foun d only scattered The through many . object was to give all that was necessary fo r a thorough comprehension o f the principles involve d and an intelligent understanding o f the

o f A kn o le method applying them . c w dg ments have been made as far as the r o f f but sou ce in ormation was known , as the lectures were prepared during o s o f i d dd and ends time , and mod fie 111 f e is rom y ar to year, it possible that proper credit has not been given in all

o f for cases . The demonstrations the ri ul fo r mula are o ginal , but the res ting mula is left in that form which the best authorities consider most desirable ; the

fo r f u authority the orm la is always given . The attempt has been made to point out

e o f all the sourc s error , and to give data showing the degree o f accuracy attain able by each method . The author trusts that this little vol ume will be found to contain all o f the essential principles and facts respe cting

s o f an d tho e subjects which it treats , hopes that it may be s erviceable to those who desire an understanding o f those

e subj cts .

U E S Y O F ILLI O IS NIV R IT N ,

No v. 6 , 1 886 . LE V E LIN G ;

ME TRIC TRIGOIVOMETR/C AND SPIRIT. BARO , ,

H e o f e § 1 . ypsom try is that branch g o desy which treats o f the measurement o f e e e e efe heights , ith r absolut wh n r rred to

the — e e e e e e sea l v l , or r lativ b tw en any two ’ he f points on t earth s sur ace . There are ' three principal an d in depen den t methods us e in . The first depends upon the law o f the decrease o f pressure o f the atm os here e e o f hi p with an incr as altitude . T s

e e e e m thod mploys the barom t r, and may

e e e be call d barom tric leveling . The s c ond depends upon the measurement o f the vertical angle and the horizontal

i r distance . It employs an angle ns t u

e m nt , the horizontal distance usually being given by triangulation ; the eleva

‘ tion is then determine d from the known o f ri parts a triangle , hence the name t gono 5 6

s metric leveling . The third con ists in measuring the distance o f two points above or below a horizontal line . This is o i rd nary leveling , in which a leveling in strument gives a visual horizontal line . Notice that the second is the only one applicable when one or both stations is e e d inaccessible . Th s three metho s will be treated separately in succeeding chap

e t rs . 2 § . In a geodesic survey conducted to

the o f the e determine size and figure arth , u the vertical element is req ired , although it is not nearly as important as the hori

al o r l o f z o nt . F example , the profi e the base must be determined so that the measurement may be reduced to a level - line , and its elevation above the sea level mus t be known , that the measurement may be reduced to the level o f the sea ; ul in planning the triang ation , at least the approximate difference of level o f the vertices o f the triangles is required to de termine the height to which the signals mus t be elevated that they may be visible

o s fr m the other station . 7

W o f the hen the object survey is a map , the vertical element is more important ; if the map is to serve as a basis o f a geologi

o r cal topographical survey , the vertical element is equally as important as the

e horizontal elem nt , or perhaps more so . If the map is to be useful in the prelim in ar fo r y examination railroads , canals , river improvements , etc . , the vertical

he element becomes t most important .

Of the e c o - 3 . thre ordinates necessary

e e e e e — 1 e to compl t ly d t rmin a point , v rti 2 cal distance ; , horizontal distance ; and

3 — un cer , direction there is the greatest tainty in the results fo r the vertical distance . It is only very recently that leveling has been done with an accuracy that would compare favorably with other

O e geodesic p rations . This is partly due to the fact that early geodesic Operations were carried on fo r scientific objects which did not involve the vertical ele ment , and partly to the natural diffi cul t i l s ies , whch wi l be discu sed presently . C HAPTER I .

E E N W M U B M L V LI G ITH ERC RIAL ARO ETER .

4 aro etr c Levelin in e e . B m i G n ral § g . The difference in height o f two p laces may be determined by finding the differ ence o f their depths below the top o f the

e e o f the atmosph r . The height atmos phore above any point is determined by i weighing it This is done by trying how high a column o f mercury or other liquid the o f column air above it will balance , or by finding the pressur e it will exert against an elastic box containing a

e the e vacuum , or by obs rving t mperature

i l i e at wh ch a iquid boils , , by observing the temperature at which the pressure o f the atmosphere just balances the tension o f he e e t vapor . This giv s ris to three

ff e e i slightly di er nt m thods , accord ng to whether the ins trument is a mercurial

‘ e e h barom t r , an aneroid , or a t ermo

e e o - barom t r or b iling point apparatus . Barometric l eveling is specially adapted to fin ding the difference o f level between 9 points at cons iderable horizontal or ver

e e tical distance apart . Und r thes con ditions the e it is most sp edy , but the

e of o f the o f l ast accurate , any methods

e leveling . It is very valuabl in making geographical surveys o f large areas fo r determ ining the elevation o f stations to be

he e o ccupied by t topograph r . It is also well suited to making a reconnois sance fo r fo r e of ul a railroad , or a schem triang a tion . T 1 THE I M AR . . S U N TR ENT .

escri tion — are § 5 . D p There two kinds o f e e the e m rcurial baromet rs , cist rn and the s yphon ; the former is the best and

e e fo r e most r liabl hypsom trical purpo ses . The general form o f the cistern baro m e ter ee i 1 n ds no description ; F g. shows the cistern and details at the lowe r en d “ O f is the instrument . The cistern made o f a e F i o up a gl ss cylind r , wh ch all ws the

rf o f e su ace the mercury to be se n , and a G o f top plate , through the neck which e t to the barom ter tube passes , and which it is fastened by a piece o f kid leather , making a strong but flexible j oint .

1 1

To this plate , also , is attached a small h the i o f hi ivory point , extrem ty w ch marks the comm encement or zero of the i i e . scal above The lower part , conta n ng

the the o f ba mercury , in which end the

ro m eter e t e f o f tub is plung d , is ormed t i e e f two par s j, h ld togeth r by our l m screws and two divided rings . To the lower piece j is fastened the flexible

N e o f e e f bag , mad ki d l ath r , urnished in

the e e k e middl with a sock t , which r sts

en d o f the s on the adj u ting screw 0 . e w the F Thes parts , ith glass cylinder , are clamped to the flange B by means of four large s crews P and the ring R ; on R re the S the ring sc ws cap , which covers the e o f e low r parts the cist rn , and sup ports at the en d o f the adjusting screw G i k o f 0 . , , j, and are boxwood ; the o o f a G ther parts br ss or erman silver . The screw 0 serves to adjust the mercury

the n to ivory point , and also , by raisi g the e fi ll bag , so as to complet ly the cistern and

r tube with mercu y , to put the instrument in condition fo r

m Mi O L s s . C . . S ith onian , Vol I 1 2

6 Fillin e — I is no . th Barom eter t § g . e fi ll slight matt r to properly a barometer . f It can best be done by the manu acturer , who has all the facilities ; but as it is sometimes necessary for the observer to l f l hi . refi l it , the ol owing nts are given Tubes require refilling owing to the breaking o f the glass or to the entrance of o f a bubble air. The mercury should be chemically pure and free from oxide ; otherwis e it will adhere to the glass and tarnish . More

e if e o f ov r , it is not pur , the height the barometric colunm will not be correct ; only mercury should be us ed which has

o r been purified by distillation . F the

e ul the b st res ts , mercury should be boile d in the tube to expel moistur e and but n ot air ; this can always be done , and fair results can be Obtained without boil f of ing . The ollowing description the method of filling is from Smithsonian R 1 859 440 reco m eport , , page , and is mended by Williamson (p . “ be The glass tube , which should O clean and dry , must have its pen end

1 4

i the several sides , when it w ll absorb

e o f n o thos minute portions air , w greatly e xpande d from removed atmospheric

r n pressure , that were not d aw at the first e fe f gath ring . The per ct reedom from air is easily recognized by the sharp co n cus sions with which the c olunm beats agains t i the sealed end , when , w th a large vacuum

e is bubble, the horizontally h ld tube ” s lightly moved . “ A barometer so prepared will prob ably read lower by a few thousandths

a if e th n the tube had been boil d , but in a stationary barometer its error will prob ably not soon change , and carrying on horseback will be apt to improve it rather than otherwise , as it is then carried with e the cistern upp rmost , and the bubbles ” the e If will be jolted toward op n end . possible it should be compared with a a stand rd barometer .

7. o § To fill a tube by b iling , an - m is alcohol la p needed , although it can fi re be done over a charcoal . The lamp to fi ll being filled and put in order, begin

the B me er . 140. Williamson on aro t , p 1 5 the tube by pouring in through the funnel as much warm mercury as will occupy fi ve e e about inch s ; then , holding the tub with both hands above the mercury , f the heat it gently , and let it boil rom surface o f the mercury downward to the o f the end tube , and then back again , o f chasing all o f the bubbles air upward .

A e i the little practic will make th s easy , tube being held a little inclined from the horizontal , and constantly and rapidly

e e so revolved , always in the sam dir ction , that every portion of the metal may be

an x o f r heated gradually d unif rmly. A te

i has e e co Ol s uf th s be n don , let the tube fi cien tly to admit o f its being held by the gloved hand , and then pour in enough warm mercury to occupy s everal inches

' o f e m a n o w be more the tub , which y s e held with both hand , one abov and the

e Af r other below the h ated portion. te

i n f e f re bo li g this thoroughly r e rom air , peat the same operation with more m er c l the ury added , until the tube is fi led to

‘ c are an d end . With practice the mer cury may be boiled entirely free from air 1 6 up to within an inch or less o f the end o f fi l wa ma the tube . A tube led in this y y h e f ave , in every r spect , as per ect a vacuum as one prepared by a professional ” i - e nstrument mak r . O 8 . In extended barometric perations i n the e o f e fi ld , a supply xtra tubes is

is carried , to be used in case a tube l broken . These tubes shou d be drawn o ut -s o as to be a little longer than they are required to be when fitted into the barometer . The open end should be cut o ff to such a length that it shall always be immers ed and yet not interfere with

e o f O he e the ris the lower part f t cist rn . When the instrument is finally put o f together, the cork in the upper end the brass case should be adjusted so as to hold the closed en d o f the tube fi rm l * y.

a i h r e r — It fre le n n t e a om te . § 9. C g B quently happens that the mercury in the c istern becomes so dirty that the ivory

e e point , or its refl ction in the m rcury , c an no longer be Seen ; this often occurs

m s 1 38 . Willia on , p . 1 7 even though the barometer be in good con in dition in every other respect . The strument can be taken apart and cleaned with safety and without changing in the slightest degree the zero o f the ins tr u * E t s O ment . very hing u ed in the pera o tion must be clean and dry . Avoid bl w

o f ts re ing upon any the par , as the moistu from the breath is injurious . “ Screw up the adjusting screw at the bottom un til the mercury entirely fills

f in the tube , care ully invert , place the e ml r strum nt fir y in an up ight position , uns crew and take Off the brass casin g which encloses the wooden and leather o f e R s parts the cist rn . emove the screw and lift o ff the upper wooden piece to which the bag is attached ; the mercury e B i will then be expos d . y then inclin ng the of the instrument a little , a portion

' mercury in the cistern may be poured out into a clean vessel at han d to receive o f it , when the end the tube will be ex

~ e posed . This is to b closed by the ns be gloved hand , wWhen the i trument can flliam o n 1 3 p , p . 6 . 1 8

i the and nverted , cistern emptied , the tube brought again to the upright posi

G e e tion . r at care must be tak n not to permit any mercury to pass out o f the

e f s tube . The long scr ws which a ten the glass portion o f the cistern to the other

e be o ff s parts can th n taken , the variou parts wiped with a clean cloth or hand kerchief and restored to their former ” position . “ If the e e old m rcury is mer ly dusty , or

the e the i dimmed by oxid , clean ng may be effe cte d by strai ni ng it through c f hamois leather , or through a unnel

the en d o f e with a capillary hole at , a siz to admit o f the passage o f but a small thread

O f f nn the metal . Such a u el is con ven ien l o f - s t y made letter paper . The du t will adhere to the skin or paper and the filtered mercury will present a clean and

If im u bright appearance . chemically

ul e f pure , it sho d be r jected , and resh , r c lean mercu y used . With such clean

‘ mercury the cistern should be filled as

e f ll n arly u as possible , the wooden por

' ti ons put together and securely fastened 1 9

s by the screws and clamps , the bras casing s e the e cr wed on , and scr w at its end

e s cr wed up . The instrument can then

e s be inverted , hung up , and r adju ted . The tube and its contents having been

is e und turbed , the instrument should r ad f ” the same as be ore . With the instrument before the Opera tor e l n e , th se instructions are easi y u d r If ee s tood . a little mercury has b n lost d O uring the peration , and there is none at hand to replace it , no serious harm has

ee e if O b n don ; but much is lost , the pen e n d o f the tube may become expos e d in

e the e a inv rting instrum nt , in which c se air hi a us may enter . In t s c se , as in ing for s and caring any in trument , a little care and a thoughtful inspection o f the method of cons truction is worth more

e e than any writt n d scription .

1 Tra or i the Barom eter 0 . ns t n § p g . s e e In tran porting a barom t r, even a be e cross a room , it should ,screw d up and carried with its cistern uppermost .

For e traveling , it should be provid d with

i m s . 1 37 . Will a on , p 20

a wooden and leather case . In steam boats or railroads it should be hun g up by a hook in the stateroom or car . In wheeled vehicles it should be carried by hand , supported by a strap over the r s houlder, or held up ight between the legs ; but it should not be allowed to of for rest on the floor the carriage , a , If sudden jolt might break the tube . carried on horseback it should be strapped ul o f over the sho der the rider, where it

r nl is not likely to be inju ed , u ess the animal is subject to a sudden change o f ab d gait . When out to be used it shoul e f a i be tak n rom its c se , wh le screwed up ,

e e gently inv rt d and hung up , when it can

e e W be unscr w d . hile it has its cistern f ll uppermost the tube is u , is one solid

of mass metal and glass , and not easily

r inju ed ; but when hung up , a sudden jolt might send a bubble of air into the

o vacuum at the upper end f the tube , and the instrument would be useless un ” til repaired .

l m s 1 34 . Wi lia on , p .

22

the ta the b arome ing final adjustment , p

e de ter a littl just above the cistern , to stroy the adhesion o f the metal to the glass . Complete the contact of the mer e cury and the ivory point , at the same tim being certain that the barometer hangs f ee i e r ly , . . , vertically . Next tap the barometer gently in the neighborhood of the top o f the mercury column to destroy the adhesion o f the

e e m rcury ; this is very important, sinc raising or lowering the mercury in the previous Operation materially affects the f o f e f e e orm the upp r sur ac . Then tak hOld lightly o f the brass casing o f the

e e barom ter, not near the attach d ther m o m eter e e t , so as not to unn c ssarily hea e e the s e e ith r ca or the th rmometer, and by means o f the mill-head screw near the l o f the e f midd e tub , bring the ront and back edge o f the vernier into the same horizontal plane with the top o f the m er the s cury in tube , ju t touching it and no more , and then remove the hand . M e e e if ov the y about , and , in any posi o f l be n tion , a line ight can seen betwee 23

he i t mercury and the vern er, the latter must be moved down a little ; if there is no

e o f e ob lin light , but a large Spac is s o nie ured , the ver r must be moved up a o f colunm little . As the top the is more o r is less convex, when the adj ustment c orrectly made a small place is obscured in the center, when the light is seen on e e ither sid . Finally having adjusted the instrument

as e e . On above , it may be r ad at l isure the best barometers the scale is usually e f- s divided to inches , t nths , and hal tenth the vernier reads to one twenty-fi fth 1 f- e — - of hal t nths (31 5 X 00 5) or two thou s andths

ART 2 THE . . THEORY .

MM O F M A . CO ON R STATICAL OR ULA .

12 Fundam ental Relations — o § . Supp se B Fi 2 e A and , g. , to repr sent two stations , an d that it is required to determine the

vertical distance between them . A and B are not necessarily in the same vertical Le . t C B line represent any point in A , ' an d D s e e a point a small di tanc b low 0 . 24

Suppos e the pressure per square inch at D P iff to be represented by , and the d erence C D in pressure between and by dP. Let a= the o f a cubic inch of air -A

B

FIG 2 . .

s of e under the condition pressure , temp ra i e - D X = . C ture , etc , ex sting betwe n and ; Of B fe the elevation A above , in et . It is clear that the increas e in pressure from C to D is e qual to weight Of a col Um n of air between C and D whose cross

in is 1 . section sq . ; or, adx= dP ( 1 )

1 If a = the o f § 3 . 0 weight a cubic inch - ° of dry air at the sea level in latitude 45

“ at the freez ing poin t when the barom e 25

P = ter stands at inches , and o the e i a pressur under wh ch o is determined ,

’ ‘ then by Boyle and Marrio tt s law

aza z s P o P a= a 0 (2) P H 0 If H e and 0 repres nt the heights o f the barometer assuming the temperatures to be the same , corresponding to the e P P = pressur s and 0 and m the weight of

o f e a cubic inch mercury , th n P Hm Hm 3 Hm 2 U 9 . 2 P, e 9 m P a Hm O 0 (4) 2 2 9 . 9 m

The weight o f a cubic inch o f air fo r any other temperature t is a H 1 o m m + ct) in which a is the coefficient O f expansion Of For 5 5 air . any latitude 9 , ( ) becomes Hm 1 1 aO

2 2m 1 cl 1 cos . 2 5 9 . 9 ( + ) ( q) e 6 v ry nearly . ( ) 2 6

4 f a f 1 . Substituting the value o rom

6 d fo r dP we ( ) in and , have d I dz: 2 2 1 cl 1 2 ‘ - 99 0 4 ) ( + cos . 15) 3 g %1 O ’ which integrated between the limits H H f o and ,, the heights the barometer at

B e A and respectiv ly , gives

( 1 + 0 0260 cos . 0 N log 3 (7)

Assuming the mean Of the temperatur e o f the air at A and B to be the mean tem

erature Of the air B we p between A and , may put T 14 1 t 1 ’ T , T and 2 1

being the temperatures of the air at Aan d

B M s . aking this substitution and pa s

ing to common logarithms , T T = — . 4 . 1 c 5 . 7 X ft . log gj + 3 2 (1 + cos . 96) (8) This formula includes the principal lations involved in determinin g difference 27 '

of al; height with the barometer . The fin formula to be used in practice has been; given differently by different investiga

o i f r t rs , accord ng to the values chosen o v f the constants , to the indi idual pre er fo r f ence one orm over another , and to

o f f the degree refinement desired . A ew Of these sp ecial fo rmulas will now be con i red s de briefly .

The ons nt — 15. C ta a The value of

— as the term 21, generally known

' ef e d the barometric co fici nt , will depen upon whose values of the den s ities o f air

r B r and mercury a e used . oit and A ago f * i = l0467 ba ound g , which makes the 0 f rom eter coefficient t . R ’ * i meters) . egnault s values wh ch are the most recent and probably the most

f In e accurate , give eet

R 1803 f * e ters) . aymond ( ) ound the valu of the barometric coefficient by determin ~ ing the value it should have to make the res ults by the formula agree with those" M m i l l I V . . sc e . Co . . Pt . I 9 S ith , Vol , , p . 28 f i The urnished by trigonometrical level ng. v is f alue obtained in this way t. meters) ; but even under the most f i O s avorable c rcumstances , the b erva t * few to e ions , eight in all , were too det r mine such a coefficient with sufficient ac F l in curacy . or reasons which wil appear hi hi e . t s and Chapt r IV , it is ghly probable that Raymond ’s coefficient is the least

h m o r fr as e . e accurate , although it been quently used than either o f the others .

The term 1 + is known as

0 oeffi the temperature term . is the c

o f of air to cient expansion , and is equal ° a roxi per 1 C . ; it is usually pp mated at If this value be sub s e tituted , the t mperature term becomes 2 T ( + T) 1 1 + fo r centigrade degrees ; if 1000 ’ T T F e is I and are given in ahr . d grees , it easily seen that the temperature term T + T’— 64 becomes 1 + , 900

R e 2 C . . 1 88 1 . 35 . U . S . G port , . p

30

= 1 X 60 58 . 6 1 2 H . 1 ( + cos q5) (9 10 ) g' ’ H X + 52252 20886860

In which X is in feet and the tempera tures in Fahrenheit degrees ; X in the last term is the value Of the precedin g of h f part t e ormula . Since the entire correction for the v ri o f a ation gravity is always quite small , and since at best the barometer can be r o f ead only to thousandths an inch ,

1 f o f which corresponds to about 0 t . alti e tude , the latitude term and also the t rm fo r variation o f gravity with the altitude may be neglected without materially af

in of Fa fect g the accuracy the results . r ther on it will be shown that the appear ance of extreme accuracy by retainin g these terms can be regarded only as a mathematical illusion , inapplicable to any

e. f real stat c practice . ’ 1 Babin et s Formulas — he f 7 . T ollow ing formula by Babin et has no term fo r

ta ’ L . . G m hs n Misc el. CO s S it o ian . Vol I , Part IV ( uyot

l e o . Co l cti n ) , p 31

f the variation o gravity . It is sometimes * c laimed that the barometric coefficient

‘ was adjus ted to meet this correction ; but from the nature of the case this can not be

e fo r n tru , except some assumed mea . H f is owever , notice that the coe ficient

a e e i see 1 l rg r than any pr v ously given , § 5.

f fo r X fee F The ormula is , in t , and ahren heit degrees , (T’+ T — 64)

H' H If and , do not greatly difi er it can readily be found that HH— H’ N , , log . H’

Making this substitution in (10) gives ’ Babinet s approximate formula H— H’ — , 64) X ft . 1 + 900 (11)

The error involved in the above for m ula is inappreciable for elevations less f t ” than ee . T

. . 9 . . . 68 . Ibid , p 7 Ibid , p 32

* The following is ess entially the same ’ as Babinet s approximate formula except the form o f the temperature term

H— H’ T + T’ 1 10 X 54500 J , HH' r ,+ 900 x 1 ft 12 4 4 0 . ( ) 200

The last two terms show the degree Of n relia ce to be placed upon the result .

orrec 18 . C tion for Temperature of — Barom eter In e it all that has pr ceded , has been assum ed that the two barom e ters were at the same temperature , which assumption will rarely or never be true .

f o f et There ore the heights the barom er , before being ins erted in the precedin g f ormulas , must be reduced to the corre spo n din g heights which they would have e at a common t mperature , or a term must be included in the formulas them selves to correct fo r the difference in tem era r of B p tu e the barometers . oth meth ods are employed . ° x n i n f f r 1 F The e pa s o o mercury o . is

- Re 1 . 352 3 . C . . 8 76 see so U . S G port , , pp ; al ’ e . 1 5 1 3d Lee s Tabl , p ( 33

of o f and that brass , which l the scales are general y made , is

1 04 iff e ex an ; the d erenc , the relative p

o f is Fo r c en sion mercury , the tigrade scale this difference is H ' ence if h represents the height o f the b arometer at the upper station , reduced ’ r o f t t to the temperatu e the lower , and , , the temperature of the barometers at the e upper and low r stations respectively , we have h’ H’[1 — d(l’ ( 13) in which d stands fo r one o f the above diff s i o f erence , accord ng to the kind thermometer used .

1 O 9. Ins tead f reducing one barom e of e ter to the temperature the oth r , both may be reduced to any other tem perature assumed as a standard; the freezing point o f water is generally chosen . E 13 e quation ( ) is still applicabl , pro i s as e vded t, be con idered r presenting i i 2° F the temperature o f melt ng c e (3 . or ’ and t the reading of the attached thermometer The formula fo r redue tion must now be applied to both read 34

: i— o f e N ngs the baromet r . umerous tables have bee n compute d fo r facilitating this — G ’ C 3d reduction see uyot s ollection , E i 61— 127 D G . G d tion , roup C , pp ; roup , 4 L ’ E 30 6 53 . e d d 3 . pp, , , , etc e s Tables , , 152— f h . 9 W o t e pp ; illiamson on the Use B 1 — 4 f i e . 6 o . aromet r, pp the Append x , etc

2 fo r ff 0. The correction the di erence in temperature of the barometers may be

’ made by ins erting a term 1n the general f n ece s ormula Thus , in it is only H’ — ’ sary to multiply by [1 d(t to reduce H’ to the corresponding height at

e o f the temp rature the lower barometer . ’ M B e s aking this correction , using abin t barometric coefficient 1 7) and approx im in d 001 e at 0 000896 . 0 t g ( ) at , we g ’ * Bailey s Formula. T' T 4 X f 0346 + l 6 t . 6 log gi 1 + 900 2 1 + cos . 95 — t 1 ,) (14) —In 1 c n Hmidit . 2 C orre tio or u § . f y de ducin e e f was g the pr c ding ormula , it as sum ed that the atmosphere was c Om

’ D 9 s C le . 6 . Guyot ol ction , , p 35 posed exclusively o f dry air ; really it is r of a mixtu e air (oxygen and nitrogen) ,

c ni an d . arbo c acid , watery vapor The carbonic acid is very small and nearly c onstant , and hence it need not be con s idered here ; but the watery vapor is If both large and variable . dry air and aqueous vapor had even nearly the same d s en ity under the same conditions , the presence o f the latter would not affect the problem ; but watery vapor is only fi ve e ns ighths as de e as dry air , and the

a o f Of Of at weight , , a unit volume the m o sphere will depe nd upon the relative a a A mount of vapor which it cont ins . c c urate hypsometry accordingly demands that some account shall be taken o f the a e o f qu ous contents the atmosphere , and a humidity term has been included in f many barometric ormulas . The introduction o f a humidity term in the barometric formula requires that the hygrometric state o f the air column s O hall be known . Accordingly an bserva tion with the hygromete r is made at each s Fo r i the wet u tation . th s purpose b lb 3 6 hygrometer or psychrometer is generally

ef o f e u pr erred , because its great r acc racy and convenience ; knowing the readings o f the wet and dry bulb thermometers , the barometric pressure due to the aque ous vapor in the air may be determined * f of rom tables , which are the results experiments . The observed heights o f the barometer may be corrected fo r the pressure of the aqueous vapor before sub stituting them in the formula ; or the Observed heights may be used un co r rected e be , and the resulting altitud mul tiplied by afacto r - to correc t fo r the hu

‘ midity . The latter method seems to be f generally pre erred .

e e s e e In a very g n ral sen e , in t mp rate climates near the s ea—level the amount o f vapor in the atmosphere is from to o f - an inch , or about one hundredth of the whole 22 Be e § . ssel was the first to propos the introduction of a correction fo r the ’ e f Plan am o ur s fo r ffect o moisture . t

l m s the B m e e e C the e Wil ia on on aro t r, Tabl , of App n ’ - — s COL B . 4 6 72 . 1 02 6 . Guyot , Group , pp , pp

38 in the stratum o f air next to the surface o f i the earth , wh ch probably contains the of i greatest amount moisture , and wh ch is therefore least representative of the

e ns vertical column betwe n the two statio . if At any rate , the gain , there is any, is not sufficient to compensate for the extra trouble in making the observations and the undesirable complication of the fo r mula , The question of the desirability of ap plying a correction for the hygrometric state Of the atmosphere is so intimately connected with the phase discus sed in

i B o f i r div sion th s a ticle , which imme diatel f f y ollows , that nothing arther need be said here . 24 onclusion — i . C The preced ng do not comprise all the formulas which have

for in been proposed barometric level g , but include the more common ones , and illustrate all the principles involved , ex B cept those discussed in division . Some of f s im e the omitted ormula are approx at , some have empirically determined pres ff th limi u . O e s re coe icients , etc wing to 3 9

tations discussed in the next division , it matte rs comparatively little which o f the generally recognized barometric formulas is used .

M F B D N C O RM . . Y A I AL ULAS

De ects o Statical Form ulas - 25 . f f All the formulas referred to above are depend ent upon the assumption that the air is in f If a state o statical equilibrium . a con dition o f statical equilibrium were possi ble , we might suppose that the whole atmosphere was arranged in a system o f

o f. be horizontal layers , each which would denser than the o n e above it and rarer

o n e e e f than the b low, each b ing uni orm

e e mi throughout in t mp rature and hu dity . The temp erature and hum idity might f if vary rom stratum to stratum un ormly , or according to some more complicated law . The fundamental assumption in deduc

o f f 1 ing the preceding class ormulas is : , that a difference o f pressure is due only ff of 2 to a di erence elevation ; , that the temperatur e o f the air varies uniformly 40 f 3 rom one station to the other ; and , that the temperature of the air between the two stations is the same as that of the vertical column between the horizontal

e of in r u plan s the two points . The t o d c tion of a correction fo r humidity involves essentially the same assumptions as the

e e temperatur t rm . 2 f 6 . The air is never in a state o stat

e uh ical equilibrium , but is p rpetually der o in o f g g local changes pressure , tem

r r Fo r e atu e . p , and humidity example ,

the of the sun , which is ultimate source

s e all di turbanc s , shines only by day . While it shines a certain amount o f heat is imparted to the whole atmosphere , but a much higher temp erature is given to the t ground , and is communica ed to the

o f contiguous layer air . At night the at m os here di p loses heat by ra ation to space , but the ground loses it still more rapidly and imparts its low temperature to the f e o . low st stratum air The lower strata , t e ef h r ore , have exceptional warmth by da y and exceptional coolness by night . If the air is moist it intercepts a greate r 4 1

Of if quantity solar heat than it is dry , so

e that a l ss quantity reaches the ground , while at night atmospheric moisture f checks radiation rom the ground . The ’ power Of the earth s surface to receive or sto re or part with heat varies with its N character . aked rocks and cultivated

s f e fields , bare earth and gra s , or st and snow are affected very differently by the

o f sun rt heat rays the , and exe equally

e diverse influ nces on the adj acent air , so that one tract Of land is often in a con dition to heat the air while an adj acent ’ e tract is cooling it . Th n , too , the sun s heat is un equally distribute d through the year ; outside the tropics there is a pro gressive accumulation o f heat through summer and a progressive loss through winter . The ocean undergoes less change o f temperature than the land , and its rate

Of fre change is slower , so that there is quent , and indeed almost continuous , contrast Of condition between it and the

As Of contiguous land . a result all these influences , together with others that

u t o might be en mera ed , the equilibrium f the air is constantly overthrown , and the to winds , which tend readjust it , are set in motion . The temperature of the air is continu ally modified by external influenc es ; the static order Of densities is broken and currents are s et in motion ; and the cir culation and the inequalities o f tempera ture cons pire to produce inequalities o f

E o f moisture . very element equilibrium

s et e e e e is thus asid , and the air is r nd r d

e e e e het rogeneous in density , t mp ratur , ” and composition .

2 Of f 7 . A consideration these acts will show the inaccuracy and ins ufficiency o f hypsometric formulas founded upon an assumed state of static equilibrium . Some O f the defects o f statical formulas will be considered in detail before discussing fo r mulas which seek to overcome thes e diffi

c ulties .

2 radien — L B 8 . G t et C A , , and des L ign ate three barometer stations . et ’ ’ ’ B C e A , , designat points vertically above each at which the pressure is the same or — K U Re 1 8 80 1 . G . e S. e . Gilb rt , in . G ological port for 43

Th . common . e plane passing through ’ ’ ’ A B C is then a surfac e o f equal pres

If e s e o f uili sure . the air w re in a tat eq b rium be e e , it would a l v l plane , but in f e e act it will be inclin d in som direction . This inclination is called the barom etric

r g adient.

n o f n e ee n I stead co sid ring only thr poi ts , we can in imagination proj ect through the air a surface containing all points i h If wh ch have t e same pres sure . the

e e rf atmospher w re at rest , this su ace

e e e would b a horizontal plan , but und r the actual conditions it is never a plane ” Fo r an d is ever undulating . small areas h f under ordinary conditions , t is sur ace would probably not differ much from a plane . Conceive another surface passed through

' all po in ts at which the pressure differs from the prec eding one by any constant

e e il quantity . With atmosph ric qu ibrium all such s urfaces would be both level

e the and parall l , but in actual case none are e are s e l l vel and no two preci ly paral el . When widely s eparated surfaces are com 44

f ll pared , the variations rom para elism are often so great that their inclinations above the same locality have Opposite i directions . The atmospheric grad ent at the surface Of the ground may therefore differ greatly 1n amount and direction from the simultaneous gradient at a con siderable altitude above the same spot . 2 Of h § 9 . The necessity considering t e barometer gradient is apparent when it is remembered that the air is continually

Of in a state motion , as is shown in the

of e o f the e variation the h ight baromet r .

Fo r if B example , A and are two stations ,

e and the atmosphere at rest , th n the sur f o f e BC ace qual pressure , , is a horizon tal plane ; and AC is the difference o f ele vation which would be obtained by ap plying any one of the preceding baro If metric formulas . the air is not in il static equilibrium , the pressure at A w l

e e f be gr at r or less than be ore , and the surface o f equal pressure may lie above or below BC ; if the pressure at A is

f o f greater then the average , the sur ace C E equal pressure is above , say at , and

46

taneous n and alike in amou t , no error would be produced by the barometer gradient ; but these conditions are seldom

e or n ver realized .

§ 30. The variations in atmospheri c pressure , and the consequent variations of i gradient , are so complicated that it s impossible to trace the relation between cause and effect ; but there are two varia u tions that are pretty well nderstood .

On e e has a daily p riod , and is caused by the variation in the heating effect of the sun between day and night ; the second the has a yearly period , and is caused by ’ variations of the sun s heat at different

e of tim s the year .

1 Diurnal rad en -I f . t t § 3 G i t. is a ac familiar to meteorologists that the pres sure Of the air everywhere undergoes a

e daily oscillation . The gradi nt intro duc ed by this daily change is called has two diurnal gradient . The pressure maxima and two min ima which are easily

i Ne r - dist nguishable . a the sea level the barometer attains its maximum about 9 f is a A M. e or 10 . In the a ternoon th re 47

i 3 M. min mum about to 5 P . It then e 10 d i f ris s until to mi n ght , when it alls n M 4 A . again u til about . , and again rises to attain its forenoon maximum ; the day

fluctuations are the larger . The daily oscillation is subj ect to varia t tions in charac er and magnitude . The oscillation is greatest at the equator and i i e dim n sh s toward the poles , but is not the

fo r o f m l same all places the sa e atitude . Within the Uni ted States it varies between

120 o f 40 and thousandths an inch . Changes of altitude Often cause a marked variation in the amoun t and character of iff the diurnal oscillation . The d erence whi ch pertains to latitude does not mate rially affect the ordinary hypsometric ff e problem , but the di erence d pending on has i ff the altitude a very mportant e ect . 2 Annual Gr adien t — 3 . The annual progress of the sun fromtropic to tropic throws a preponderance of heat first on one side Of the equator and then on the

h o f other , w ich produces an annual cycle changes in the pressure and gives rise to what has been called the annual gradi 4 8

ents . The amount o f this variation is

in quite small near the equator, but creases rapidly toward the poles ; at the equator it rarely exceeds i Of an inch

the i per year , while in polar regions it s Often as much as 2 or 3 inches in afew ” days .

u- c - 33 . No periodi Gradien ts In ad dition to the diurnal and annual varia tions in the pressure there are others due

e e of to the same gen ral cause , the h at the sun , but so modified by the local con ditio ns— d topography , the humidity , win s , — . e storms , etc as to make it impossibl to o f discover the law their action . These non-periodic variations are much greater in amount and more rapid in their actions f than either o the others .

4 er re Tem atu radien t. § 3 . p G It has just been explained that the variations of pressure are due primarily to inequali ties o f temperature ; it will now be shown

if ffe o f e that , di rences el vations are deter

' the fo rm ulas mined by commonly used ,

’ * Wi ms . 68 . ill a on , p 49

the temperature is directly responsible for other and generally more serious They arise from the difli culty

of determining the temperature of the vertical column of air between the two stations . 50

L B Fi 4 et A and ( g. ) be two stations the difference Of elevation o f which is to be obtained from Observations of the barometer and thermom eter made at Let a e ach . it be ssumed that the pressure O bserved at B is the same as that at C

r . - B ve tically over A and on a level with . T o m f a use the statical or com on ormul s , the temperature o f the column AC must be known ; in applying these formulas it is assumed that the mean tempera ture of this column is equal to the mean of the temperatures Observed at B A and . Ho w admis sible this assumption is will appear at once when the mann er in which the air acquires and loses heat is of recalled . The body the atmosphere is heated directly by the sun and gives ff i o ts heat by radiation into space . The surface Of the earth is heated and cooled in n the same man er, but many times is more rapidly , so that by day it always Of much warmer than the body the air, i and by night it s much cooler . A layer of air next to the earth receives its warmth 51

f to rom the earth , and is thereby caused differ widely in te mperature from the f remainder o the atmosphere . Not only is the greater part of the column inac cessible to us , but that portion to which o ur observations are restricted is the por f ll ” tion least representative o a . By me asuring the difference Of eleva tion o f two points with the Spirit level m f reversing the baro eter ormula , and computing the temperature o f the air f u e column , it has been o nd that in middl latitudes the average daily range o f the temperature o f the body Of the air is . about and that o f the superficial ° ° f 10 20 the layer , is rom to near sea ° ° f 20 35 in shore , and rom to in the

ri r f te o o continents . There is a stratum f Fi o f air near the sur ace ( g. 3) which

l l r oscil ates dai y through this wide ange , while the temperature of the upper and lower portion of the column AC is re

i l n ef lat ve y co stant . Ther ore the mean o f the Observed temperatures absolutely

Re . Gilbe rt in U . S . G . S . port 52 fails to give the mean temperature o f the colum n AC as required by the fo r m ula .

Nor . § 35. does the trouble end here Whenever the ground layer is cooler

e o f e e than the air abov , it is cours h avier , and e e , like any oth r h avy fluid , it flows

w an d e e do n hill accumulat s in vall ys , f e o f orming lak s cold air . The nightly layer o f abno rm allv cool air is therefore t e e hinner on emin nces than in vall ys , and the contrast increases as the night ad

h r . W e V ances en the conditions are versed so that the ground layer is warmer

it e e than the air above , it has a t nd ncy

e e the to ris , but accomplish s change in an e e e irr gular mann r , br aking through the immediately superior layer here and there an d r isin g in streams which spread out in sheets wherever the conditions ” o f e equilibrium are r ached . Observers

l s e e d in ba loon , as th y asc nd or descen , rarely find an orderly success ion o f tem

ra ur s If ef in e t e . p , ther ore , we could s ome way determine the temperature o f s ome point in the upper portion of the

54 state are still further increased by the

of f o f laws condensation . At the sur ace the earth there is an almost continuous

o f f air passage moisture rom ground to , only a part Of the total exhalation being d returned as dew . The aily circulation incited by the heat o f the sun carri es the moistened air upward and eventually the water is returned to the earth in the form o f the ens rain or snow , but cond ation

' and succeeding precipitation are exc eed

' l W e f in . g y irregular henever, th re ore , a current of air moves upward and its

ef temperature is lowered by rar action , a point may be reached where the aecom panying vapor can no longer exist as

c e e su h , and is condens d to cloud or ev n On to rain or snow . the other hand , whenever a current Of air moves clown fo r i ward , its capacity moisture is n

- e creased , and it acquires a quasi absorb nt f power, so as to take up water rom what ” ever moist surface it touches . The irregularities of humidity are greater pro po rtionallv than the associ of the ated irregularities temperature , but 55

error in the. difference of altitude due to humidity is less than that due to tem

erature u p , because h midity is a much

f o f h so m etrlc e smaller actor yp probl ms . ’ m la —F * Ferrel s For u . e e § 37 . rr l has deduced from a consideration of dvn am

e e r f ical principl s , a barom t ic ormula i d e “ f wh ch istinctly recogniz s the de ects , as e e Of f l f discuss d abov , ormu as ounded upon a statical condition o f the atmos

here e Of p , and which indicat s a method h remedying them . Although t e formula is very ‘ c arefully and ingenious ly worked

et r o f e fo r out , y it is p obably littl use o w rdinary hypsometrical ork , since it requires observations to be made fo r a l s e ong time over a con iderabl area , to get the data by which to compute corrections fo r gradient , temperature , and humidity . Without the data fo r makin g these

e h f m r ductions , t is or ula is essentially the same and has essentially the same defects as the formulas depending upon a statical condition o f the atm o sphere fr

C Re 1 1 22 — . . . . 8 8 . 5 68 . U S S port , , pp ‘ l m e r e . W. . C . . e 1 . . R 88 1 . 243 1 F r in U S G S port , , p . 56

’ ilbert s Form ta — * G . G rt h 38 . a ilbe as developed a method fo r deter mining heights with the barometer which does

O e o f not require bs rvations the tempera.

hum idit o f Hi s ture and v the air . method requires simultaneous Obser vations of the

barometer at three stations , the verti cal distance between two o i which is known ; from the known difference be tween two o f the stations and the Obs er

vatio n s the e o f the at each , actual d nsity air can be found ; then the true density can be used to compute the differenc e o f elevation between either o f thes e s tations

and the third . The method is most accurate when the three stations are in the same vertical and when the one whose elevation is

ee ff desired lies betw n the two , the di er ence o f whose elevations is known ; the

method is applicable , but is less accurate , when the stations are not in the same is vertical , or when the one whose height sought lies either above or below the

~ other two . - e . . Re 1 88 0 1 . In U . S . G ol S port , 57

One of the distinctive characteristics o f this method is that it Observes density d e O irectly , whereas oth r methods bserve temperature and moisture only and de has duce density . The only reason which ever existed fo r measuring the tempera ture Of the air and the moisture in it has

i n i e been to ascerta n its de s tv. A s cond distinctive feature is that this method employs in its determination o f dens ity a column of air comparable in height with the one to be measured and fairly

e o f w l representativ it , hi e other methods base their diagnosis o f the column to be measured on density determinations made u close to the ground , where , as a r le , the ” conditions are not representative . It will be show n presently that this method f also has serious de ects . f fo r § 39. The ormula this method is deduced as follows :

Let L N h t , , and U represent t e al itudes o f the lower new, and upper stations re s ectivel I n u p y; let , , and represent the synchronous barometric readings at these 58

same stations corrected fo r temperature of the instrument and instrum ental er L E = ro rs . et o r the vertical base line , the known difference of altitude o f the l E = — L U . upper and ower stations , Let A= the required difference of alti N — L a= an tude , and let approximate

o f value A .

E = U — L Since , A= N— L B — A=U— N and , .

Fo r convenience refer all vertical dis as tances to the lower station an origin . If fo r the present we neglect the de crease in temperature and moisture with o f altidute an increase the , and assume that the accidental or temporary varia tions Of density due to temperature and humidity are the same in both columns , the following proportion may be made

The approximate the true height " height o f the Of the base l B base line ine , , ’ the approximate the true h ght height of the o f the new

new station station , A . 59

The approxim ate height (length) Of the l f base ine , as deduced rom the readings o f the barometer at the two stations is , — hi l . u 0 C (log . log ) in w ch is the barom

eter constant . In the same way the approximate height of the new station l— n C . . above the lower is (log log ) . ro Substituting these values , the above p

portion becomes , — B l . u C (log . log )

— C . l n A (log log . ) : .

— l . log . log o r a= B in which a is — u l . log . log

Of written instead A , owing to the neglect o f the variation o f temperature and

humidity with the altitude . 4 u 0. The preceding eq ation would give the correct result if the atmospheric f column were uni orm in temperature , and if its aqueous vapor were uniformly distributed ; but since this is never the case there must be added a term which shall take account Of the variation Of temperature and moisture with the alti

tude . 60

It is well known that in a general way the temperature and moisture decrease with the altitude ; but the exact law of this variation has not yet been discovered . f f There ore , be ore the correction to be added to equation (15) can be deter i m ned , it will be necessary to assume fo r i some law th s variation . “ If o f f s the air were uni orm den ity , and the element o f temperature were in troduced alone , the high temperatures at low altitudes would cause a dilation

e there , and the low temp ratures at high altitudes would cause a contraction , and the resulting distribution of densities would be characterized by an increase If f from below upward . the air were o uniform density and the element of vapor distribution were introduced alone , the of greater per cent . aqueous vapor (which is a rarer gas than dry air) in the lower strata , would cause them to be relatively of rare , and the resulting distribution densities would be characterized by an increase from below Con

. e ve Re 188 1 . 4 4 1 . U . S G ological Sur y port , , p

62 tween the lower and the new station is at

i he a point above t lower . The vertical 2 B — — d1 8 tan ce between these po1nts Is 2 2 The decrease of dens ity from the middle point of the colum n A to the middle point E A of u B ; the col mn is ; % that is , the B A mean density of the column A is 2D

of B greater than that the column . In deducing the first te rm (15) o f the f u proposed ormula , it was ass med that the density as far as it depended upon temperature and humidity was uniform throughout both colum n s ; but we have jus t shown that the element o f the den sity varies directly as the altitude ; con sequently a term must be added to (15)

f ' to correct or the variation . It has j ust been shown that the mean dens ity o f A is B — A of B m greater than that . The ean 2D dens ity o f B is the unit or standard d en i s s s ty con equently , to express the den ity . 63

of of B of as A in terms , the density A , assumed in must be diminished by B — A — A . 2D

F of of inally , the neglect the variation density with temperature and humidity

fo r assume too great a density column A , and since heights are inversely propor tion al to densities , that which makes the density too great makes the height too small ; therefore the height of A as de duced from (15) must be increased by the B — A quantlty A . 2D

Adding this term to (15) gives fo r the

of height A the new station ,

(10) — u D l . 2 log . log . a Since the last term is always small, fo r can be substituted A , thereby making the formula more convenient to compute .

42 If o f . the position the new station be referred to the upper station instead o f f re the lower , the above ormula will l n l main unchanged , except and , and and u will change places . 64

f The ormula is applicable , even though the new station is ' not intermediate in If height between the other two . the new station is above both the others , the B — A quantity ( ) then becomes minus , and the last term is subtracted ; if the new

e station is b low both the other two , the

m of ll u nu erators both fractions wi be min s , and the result will be the sum of the two * terms . 4 D f 3 . The quantity can be ound only m by experi ent ; to find it , observe the barometer at all three stations , determine B - D t . A and by a spirit level , and compu e The exp eriments should cover a great range o f conditions so as to secure a fair f D mean value . Un ortunately has not yet been determined from suffi ciently varied conditions . The only value known is o n e dete rmined by Gilbert from Obser vatio ns of at only two sets stations , and

' o n e Of them was not very satisfactory . In f D= 245 000 this way it was ound that , feet . The internal evidence o f the observa

2 e . 44 . Gilb rt , p 65

tions from which this value is derived is “ such that it is probable its real value will eventually be found to be somewhat smaller than the one provisionally as ” sigHned . appily , the last term is always rela tivel y small , and hence any uncertainty in the value of D will have only a small

Th n effect upon the final result . e u cer tainty in the value Of Dis the chief defect

f . in this ormula . Introducing this value

o f D 16 , ( ) becomes — n A B — l . A log . log ( ) A(m feet) — B (17) l— u log . log . — A B — log . l log . n ( A) e B + A(in met rs) — l . u log . log ( 18)

44 Reduction Tables — f . The use o all the preceding formulas is very much simplified by tables which facilitate their G ’ application . uyot s Collection (Smith sonian M 1 iscellaneous Collection , Vol . ) contains tables fo r the application of all the principal statical formulas The Ap

l e . 502 Gi b rt , p . 66

n dix of W n pe illiamson O the Use of the Barometer contains a series o f tables fo r ’ ’ La Place s form of Plantam o ur s formula R ’ ff with egnault s coe icient . The ’ s e L ame tabl s are also given in ee s Tables ,

— 4 1 2 Th . . 1 . e G pp . 8 8 U S eological Sur vey Report fo r 188 1 contains the only ’ table necessary in applying Gilbert s fo r mula . Tables are useful where a great num ber of observations are to be reduced ; but they generally contain an un n ec es

o f f sary number figures , and hold orth a show of extreme accuracy which the na ture o f the observations themselves can if no t just y. 4 5 . In the succeeding article it will be shown that statical formulas are gen erally applied in such a way as to largely eliminate the defects referred to in this section . It is impossible to completely e liminate the errors due to gradient , tem ff erature . n e p , etc , and , co sequently , di r

‘ ence o f elevation cannot be determined with precision by means of the barometer . 67

A T P E R HE R C C . 3 . T . A TI

‘ 4 mm on M od — R . o h § 6 C i et . esults may

e be obtain d by using only one barometer ,

is e f which carri d rom station to station , o n e or more Observations being made at e ach station ; but results Obtained in such a manner would be only rude appro xim a w f s o i . tion , o ing to errors grad ent The greate r the distance between the two h h D po ints t e greater t e error . istant stations are sometimes connected by in term ediate ones . The errors due to change Of gradient are partially eliminated by making simul

aneo us O t bservations at the two stations . If the phas e and the amplitude Of the

e e variation wer the sam at both stations , which probably seldom or never occurs , simultaneous Observations would give re s ults e e e o f i Of e ind p nd nt th s class rrors . E rrors due to gradient are still further reduced by making a number o f simul taneo us observations and using the mean ; this eliminates only the variable element and fails to take account of permanent gradient . 68

It is often recomm ended that the ob s ervations be made at certain hours o f the

h e day , at w ich tim it is supposed the diurnal and annual gradients are zero . These times can only be determined from

experiment , and will vary with the state

o f the the atmosphere , the season , local

. . S. ity , the elevation , etc The U Coast

S followm e urvey recommend the g tim s ,

subject to the preceding limitations . They were probably deduced fo r the mid

dle Atlantic coast . The hours refer to of the middle the month , other times determ m ed are to be by interpolation .

1 P M . .

1 M an A. e rua 0 . d 4 F b ry . 8 l ‘ U 6 7 May 7 7

Jun e . July Augus t 7

e m e 8 6 Sept b r . Oc to b er 1 0 o em e N v b r . m at n o tim e De e e . . . c b r .

R e 1 876 349 port , . p . .

70 the Obj ect of both series being to ascertain the nature of the diurnal variation of u pressure and temperat re .

The barometric readings at the . base s fo r of station , corrected temperature the instruments , are plotted upon ruled paper so as to exhibit their curve , and all read ings shown by ins pection to be influenced by abrupt and violent atmospheric dis turban ces - s , such as thunder storm , are discarded , their places being filled by F o interpolations . rom the corrected b servations s co rrec at the base station , a tion is deduced , which , being applied to the several barometric readings , reduce them to the daily mean ; applying this correction eliminates at least part of the effect of diurnal gradient . Instead of determining the temperature of the air column from the temperature

m o f at the ti e observing , the mean tem perature of the day is used ; this can be quite accurately determined at” tiie base ' a roxim atel known stations , but is only pp y N at the other stations . otice that the mean of the daily means will not be the 71 mean temperature of the vertical air

column . The difference of altitude can then be computed from the reduced barometric e readings and the mean daily t mperature , by using any of the statical formulas ; Williamson hims elf used his translation ’ O f Plantam our s formula ’ “ hitne s Method — F 49 . W ob § y rom s ervatio ns made in connection with the S of f Geological urvey Cali ornia , a series Of corrections were deduced for reducin g the barometric readings made at differ

o f o f ff ent hours the day , the di erent days

O f ff s fo r ff the di erent month , and the di er e nt altitudes to the daily mean fo r the year . These corrections can only be used in the neighborhood in which the observa

s tion on which they were based were made . Similar tables made for different cliin ates ff f ’ would di er materially rom each other . Fo r tables constructed upon this principle for of G P the climates ermany , hiladelphia , ’ G se and reenwich , respectively , e Guyot s D d E G 3 d . . 80 8 1 Collection , roup , , pp , , 4 93 , 9 . 72

’ Plan m our s M — 50. ta ethod In h . t e

ri of S P a hypsomet c survey witzerland , l n tam our made Simultaneous Observations of s the barometer , thermometer , and p y hrom ter B c e G St . at eneva , ernard , and at the station whose height was to be deter ff mined . The approximate di erence of altitude between the new station and G B St . eneva , and between it and ernard , ’ were computed by Plantam our s formula 22) the difference o f elevation between B Geneva and St . ernard was also com u diffi eren ce o f p ted . The computed ele B vation between Geneva and St . ernard compared with the actual difference of

e the s - e altitud , as determined by pirit l vel , gave a correction to be applied to the compute d differences fo r the new sta

s o f tions . The ingeniou details the com putation are to o complex to be described here . Marshall and Ruhlm ann applied meth ods somewhat similar to the above . ’ —

. Me hod 1 Gi lbert s t . 5 . This method ,

’ ve . . . 522. Whee ler s Geological Sur y , Vol II , p 73 which has already incidentally been fully described , is somewhat Similar to the ff f indeterm in above , but di ers rom them ing the height o f the new station by the f sole means O the observed pressures . A comparison between it and the several

' other methods seems to prove that it is * the most accurate . 2 f t of 5 . Un ortuna ely all methods elim in atin g gradients involve considerable s time and expen e , and even then do not thoroughly accomplish the desired end , all of which Shows that when great accu racy is desired the barometer Should be ff dispensed with altogether , and the di er enc e o f elevation dete rmined by some

other means . rces o E rro — Th ou . e § 53 . S f r principal

Of of sources error , as well as the means

m ‘ e eli inating them , hav already inciden i tally been d scussed , and need only to be

referred to here . Ins trum ental Errors as 1 . , such index

m f of the error, i per ection scale , imper

’ e s . . . . R e 1 88 1 Gilb rt U S G S port , 74

f e for of e e t adjustment capillarity the tub , i mpure mercury , and errors in the at h tac ed thermometer . The first is usually eliminated by an u of of adj stment the zero the scale , and with a good instrument the others would

be inappreciable . E rrors o Observa ion 2 . t as f , inaccuracy Of making contact between the ivory

i of po nt and the mercury , inaccuracy the lf reading itse , and the inaccuracy in deter

i o f m ning the temperature the barometer . Gilbert * from a comparison of 360 pairs o f observations made by the Signa Service and the Geological Survey found the average error of Observation to be a - trifle less than three thousandths of an in diff ch . This erence does not involve

the personal error between two observers , hi fo r O w ch , even expert bservers , may be

nearly as much more . t Errors due to Gradient an 3 . , diurnal ,

nual , and abnormal, and those due to m temperature and hu idity . The errors

- v 1 1 542 . Re o . . Geo lo . u e 880 . p rt U S g S r y , . p ‘ . . C . 8 . Re 1870 . 79 . 1 U S port . , p 75

of this class may have ahn ost any value ; - the various methods of partially elimi w

nating them have already been discussed . d 4 E rrors due to the E ect o the in . . ff f W The wind may cause either a condensa tion or rarefaction Of the air in the room

in which the barometer is , or even in the of f i cistern the barometer itsel . Th s effect will vary with the velocity of the wi of wind , th the position the openings f O . n with re erence to the wind , etc M W o f 50 ount ashington , a wind miles 1 per hour caused the barometer to read . 3 ff of an inch too low . Its e ect will vary as f the square o the velocity . It may be n if early , not wholly, eliminated by hav

o n e ing two apertures , each on the wind ward and leeward side Of the inclosed * space . A similar effect Of the wind is caused when the ins trument is read in the imm e

' diate vicinity of any body which Ob~ F r if structs . o the wind example , the barometer is observed on the windward

' o f a u the be side mo ntain , reading will

— e . . Geo lo . ve R e 1 88 0 1 . 562 Gilb rt , U S g Sur y port , , p . 76

i if . too h gh ; on the leeward , too low The only way to avoid this difficulty is in the selection of the stations ; but it is not always possible so to avoid it . Limi — 54 . ts of Precis ion It is some “ times stated that the barometer is the most accurate instrument fo r determin ” i of nl ing d fferences level . It needs o y a moment’s reflection to see that this can f l w t not be true The ol o ing resul s , given P fe G f by ro ssor uyot , are requently quoted as showing the great accuracy Of baro metri c instruments

Mon B an c arom e te ee t l , by b r , f t s i i - eve f by p r t l l , eet o n W n to n a m Mu as i o ete fee t h g , by b r r , t; s i i t-eve eet by p r l l , f . In No a o in a a om eter 6 701 ee t rth C r l , by b r , f ; by S irit-eve p l l , feet. In Nort aro in a a o m e te ee : h C l , by b r r, f t s iri t-eve ee by p l l , f t. These results are to be considered ex c e tio n al l O p , and on y btained by num er o us repetitions in various states of the a tmosphere . The difference o f altitude computed ’ from one or even several days Observa

78

f tions is used , which was ormerly sup

e b e a a hi n pos d to tt inable by t s mea s .

. The results by the barometer were Ob tained by computing the difference o f altitude from monthly means Of the mean

f the d i O r n o a ly bse vatio s , and taking the * mean fo r the time stated .

Sa am en to an d Sum m it cr , ’ e ars o se vatio n s 24 ft in 3 , ft y b r , . . Gen e va an d St Be n ar . r d , ’ 1 2 ea s o s e vatio n s 2 6 m e t in m . e t y r b r , . Po t an an d Mo un t Was in to n r l d h g , ’ 6 e ars o s e vatio n s 3 f in 7 t. ft y b r , . V era ruz an d i o f Mexic o C C ty , ’ 1 ea s o s e va i o n s m e in m 5 t. t y r b r t , e

Fo r m r in 56 . an te est g comparison Of

s e the absolute , and al o the relativ , errors ’ o f e G r the various m thods , see ilbe t s M h C . U. S. G emoirs , apt III , in eological R 18 — 1 80 . eport , Fo r additional data concerning the ao Of l e . curacy barometric veling , see U S.

. G . S e Re 88 . 1870 . C urv y port , , p ; do , 1871 1 4— — . 5 1 5 7 . 8 6 . 355 6 7 . , p ; do , , p 7

. . C . . Re 1 88 1 . 254 U S S port , . p . 79 .

Al 57 . though the barometer can not be regarded as a hypsometric instrum ent o f et great precision , y with care it can be made to give results with sufficient accu

fo r racy reconnoissance or exploration . Fo r this purpose it is unexcelled by any um other instr ent , but this is about the o nly use o f the instrument to the engineer

ing profession .

HPT E R II C A .

E IN I T HE N E RO ID B ROME E LE V L G W TH A A T R .

1 THE O MMO N N E AR T . . C RO A ID .

Descri tion — The 58 . p aneroid barom e ter consists of a cylindrical metallic

e o f o f is box , xhausted air , the top which o f t e SO made hin corrugated m tal , elastic that it readily yieldsto alterations in the

o f pressure the atmosphere . When the

e o e e in pr ssure increases , the t p is pr ss d the wards ; when it decreases , elasticity o f the lid tends to move it in the opposite m direction . These otions are transmitted by delicate multiplyinglevers to an index

whi e S ch moves ov r a scale . A pring is 80 sometimes ins erted between the two ends Of the vacuum chamber to reinforce the of elasticity the corrugated ends . Some times the vacuous box is not entirely ex n ha sted , the object being that the enclosed “ f air may rein orce the spring , the air gaining elasticity as the spring loses , with ” n of e s i crease altitud . It is at lea t doubtful whether the Spiral spring needs a f the ssistance , and there ore whether air is o f any benefit ; and it certainly in tro duces the complications , owing to

ff o f o f e of the e ect, a change t mperature enclosed air . There are several forms o f aneroids which differ in the mechanism employed to multiply the linear motion o f the end o f the vacuous box and to convert it into Fi 5 angular motion . g. shows the mech anis m o f a common form ; the outside case and the front face of the vacuous box are removed . The instrument is graduated empiric ally by comparing its indications under different pre ssures with those of a mer curial barometer ; the scale is marked to

82 corr espond to inches of the ordinary ba; rom eter c olun m , the inches being divided

, e into tenths "and the t nths usually into f of i our parts . At the back the ns tru ment i s a little screw whi ch presses agains t of e u one end the exhaust d box , by t rn ing this screw the index can be moved ins over the scale , and the trument may

. thus be made to agree at any time with a m rcurlal standard e barometer . n 59. In many instrume ts there is an additional scale Of. altitudes in feet gen erally divided according to a table pre for P f Ai pared the purpose by ro essor ry . Such a table could be prepared by using any o f the formulae discussed in the pre ceding chapter by neglecting the c o rrec

‘ P f r Ai r f l tions . ro esso y used a ormu a similar to an d neglected the tem

' eraturet rnL Wen p e h the aneroid has a scale of elevations engraved upon its face the approximate difference of height is Obtained by subtracting the reading in feet at the lower station from that at the upper . The use o f such a scale leads only to 83

is s rough approximations , as it ba ed on the assum ption that certain differences o f pres sure correspond at all heights with if f the same d ferences o elevation . The scale o f elevations can only be correct at ul e r some partic ar t mperatu e , and hence in general a temperature correction mus t i o av be appl ed . S me makers ende or to e liminate this correction by making the “ The scale movable . movable scale is ” n u s cientific and inaccurate . The best to Of plan is dispense with scales altitude , e whether fix d or movable , and calculate

the heights . D — 60 . efects The aneroid is a very c u for S onvenient instr ment , and a tation ary instrum ent where nice readings are

ui f r not req red , it does very well ; but o accurate hypsometrical res ults it is an f I f in erior instrument . ts de ects are

1 . The elasticity Of the corrugated to p Of the vacuum chamber is affected by re eated p changes in pressure . This will i produce error in the scale read ngs .

2 . It is usually claimed that , in con sequence o f not completely exhausting 84

the s of th vacuum box , the indication e aneroid become independent o f the effect of changes of temperature of the instru

f r ment . The best that can be hoped o is that for small changes the temperature correction is less than the error of obser

f r vation . In instruments compens ated o

‘ r eff Of is temperatu e , the ect a change s ometimes the same as that in the mer curial barometer and sometimes the re

ff o f e verse . The e ect the temperatur on any particular instrument can be deter n mined o ly by trial . ff 3 . The di erent spaces on the scale are c seldom correct relative to ea h other, owing probably to errors Of observations and to ff graduation , and possibly di er en ces of temperature and changes in elas i i of f is t c t . y As a matter act , the scale f often only a Scale o equal parts . The barometer scale is more accurate than e has the elevation scale , since the latt r all the inaccuracies due to the formula i t in by which t is gradua ed , addition to those of the instrument itself Fo r accu rate workthe aneroid should have a ther

86

s is ns omewhat ; that , the i trument runs B f i . s down e ore u ng the instrument , ex perim ents should be made to dete rmine - the range of pressure to which it may be f e xposed be ore the Spring ceases to act . In case an aneroid is to be used in an e if t u levated regi on , here is a merc rial w barometer ith the party , screw up th e aneroid until the spring acts well and set

SO it by the mercurial barometer, that there Shall be a difference of say one

or two inches between them . “ f § 61 . With all these de ects a good aneroid 1s of much assistance on a survey or reconnoissance in mountainous dis ’ o f tricts , on side trips one or even several ’ days duration , when the instrument had been previously . compared with a stand ard mercurial barometer at various tem peratures and in different elevations and is proper tables of corrections made . It evidently impo rtant that there should be

a good attached thermometer . It should be compared before and after it is used

‘ see if ze in that way , to the ro has not c if hanged in the meantime , and the 8 7

agreements are satisfactory the results ” n ca be relied upon .

Formulae — F o f 62 . rom the readings diff the aneroid at two stations , the er ence of elevation may be d etermined by any o f the form ulm o f the preceding chap

fo r of a roxiinate f as ter, any the pp ormul are as accurate as the instrument . A ’ modification Of Babinet s approximate formula (1 1) is most frequently used . The following is a very common form : H’ T — 55 X X 545 1 I10 450 l 200

The last term is the supposed probable error due to the varying dens ity o f the e e t air column , and the prec ding t rm tha

f fo r due to the instrument itsel . This mula is lim ited to difference Of heights f of about eet .

2 THE GOLDSOHMIDT E ART . . N RO A ID . m 63 . The com on aneroid was in t o f P 1847 th ven ed by Vidi , aris , in , and e defects o f its complex levers have long

il m s the B m e e W lia on on aro t r.

' . . C . . Re 1 876 . 352 . 1 U S S port , , p 88

e o ni As -as b en rec g zed . early s chmidt designed a form of aneroid which

FI 6 G . .

to do away with the transmitting and multiplyin g mechanism of the Vidi 89

f F . 6 7 of on orm . igs and are two views e ’ of the latest forms of Goldschm idt s ane i F . 7 roids . g is a section through the com

FIG . 7 pound vacuum chamber ; the greater the number of boxes the larger the motion of a o f the the index . The relative position movable index a and fixed point of refer ence b is observed by the telescope L

(Fig. the dis tance being measured by 90

h M t e micrometer . The instrument is 1n 1ts very delicate indications , but is lia ble to serious disarrangement by ordinary Dff f handling . i erent manu actures have s lightly different forms o f the Goldschmidt all l type , but have essential y the same defects— are not able to stand ordinary

us e . It is doubtful if there is any advantage in an aneroid as complicated as that 6 s hown in Figs . and 7 ; it seems probable that no form can be devised which shall be both delicate in its indications and Th hi able to stand rough handling . e c ef advantage Of the common aneroid is its r po tability , combined with moderate accu r acy. The mercurial barometer and the aneroid supplement each other ; the first is delicate and the second is portable . It is doubtful if the two qualities can be com bin i u o n O ed in a S ngle instr ment, or e b tain ed more delicate or more reliable than the mercurial barometer . 91

ER CHAPT III .

E E N THE HE RMO-B ROME E L V LI G WITH T A T R .

— 4 heo . W T r e e 6 . y h n wat r is heated , the elastic force . of the vapor produced from it gradually Increases until it be comes equal to the incumbent weight Of

. e e the atmosphere Th n , the pr ssure of

e the atmospher being overcome , the steam escapes rapidly in large bubbles an d the S water boils . ince the temperature at which water boils in the open air depends upon the weight of the atmosphere col m u the wei ht of u above it , and as g the 0 atmosphere decreases with the elevation , O u it is bvio s that the higher the station , the lower the temperature at which the water will boil at that station . The tem peratur e at which water boils under differ ent pressures has been determined by ex

n perim e t. It is then only necessary to observe the temperature at each station f at which water boils , and by re erring 92

the to tables Similar to the above , find of corresponding height the barometer, from which the difference Of elevation may be computed by any of the formulae h previously given . Or t e temperature

FI G . 8 .

O l o an may be bserved at on y one p int, d by using the mean pressure at the e level , compute the absolut elevation .

‘ if eff Of r , s O , finally, the ect variation es in temperature , moisture , pr sure, etc

94 from pure wate r while boilin g under at m s h ri o p e c pressure . The general arrangement consists o f a closed vessel with a chimney with a com bination o f passageways fo r the exit of i F 8 . e . the steam , somewhat lik g The bulb of the thermometer is thus immersed

o f in a current steam . The double pas s ageway is to prevent condensation on

e o f the inn r walls the flue .

De ects — f ff is 66 . f The chie di iculty in ascertaining with the necessary accu racy the true temperature Of boilin g

I e water ; an error o f T , Of a degr e centi grade would cause an error o f 70 to 80 f fi eet in the nal result . An observation

l ' o f the in - ff o f a boil g point , di ering by T , f e degree rom the tru temperature , ought

o ne to be considered a good . The accu racy is dependent upon the accuracy and

s e e of the is sen itiv n ss thermometer, and affe cted by the quality o f the glass Of the

e e the f o thermom t r , orm and substance f the vessel containing the water , the purity of the e e wat r, the plac at which the bulb of e e the thermomet r is placed , wh ther in 95

o f the current steam or in the water, the o f e of error r ading , displacement zero point , etc . E if ven the above errors did not exist, this method would still be subj ect to all the chances of error which affect the measurements o f heights by the barom e ter . Nor is the thermo-barometer as con venien t as either the aneroid or mercurial barometer , owing to the time required to s Ob tart a fire , boil the water , make the

fo r the n servation, and wait i strument to c and f o f O ool, the di ficulty btaining pure

e u water . Also , altogeth r the apparat s makes quite a load to be carried from

place to place . S the o f ince ( invention the aneroid the method o f measuring heights by the tem perature Of boiling water has almost been abandoned . 96

E R CHAPT IV.

R O N O ME RIC E E I T IG T L V L NG .

— Princi le . r 67 . p T igonometric level ing consists in determining the difference o f level of two stations by means of the measured angle o f elevation or zenith dis tance Of one and the known horiz ontal

distance between them . The horiz ontal

distance is usually given by triangular

tion . This kind of leveling is peculiarly Suit able for finding the heights of the sta

s o f r tion a triangulation su vey , since the extra labor required to measure the n eces

i e sary vert cal angl s is but slight . Observations — 68 . The vertical angles are meas ured at the same time and with the same instrument as the horizontal

angles . The instrument should have two opposite verniers or micrometer micro scopes and a sensitive level in a plane l parallel to the vertical circle . It shou d

ef for im ver be car ully adjusted coll ation ,

98 as to obtain different conditions o f the atmosphere rather than to take any great o f s O number succes ive bservations . Re raction — S ff f 69 . f ince the e ect o atmospheric refraction is to elevate o h

ects fo r j on the horizon , a correction refraction must be added to the zenith

s un c er_ distance a measured above . The tainty as to the amount o f this correction is the great cause o f inaccurate work in

i R f is trigonometrical level ng . e raction so erratic in its character that no satis factory method has yet been devised for

i The s determin ng it . be t that can be done is to Observe only when the refrac ff tion has its least e e ct .

f e of There or , since the accuracy trig onom etrical leveling is limited by our knowledge Of the laws of atmospheric

f e re raction , it will be n cessary to investi gate that branch before proceeding with

tlie general subject .

ART . O E IE F E . C C O R R C I FFI NT F A TION .

De nition — o f r fr 70 . fi The angle e ac f tion divided by the arc Of the earth s 99

f e the circum erence , int rcepted between

e is observer and the station observ d , ‘ * called the coefi ici en t of refraction . That if O= n o f the is , the angle at the ce ter F = earth subtended by the two stations , the refraction angle and m = the co effi F - of f m . C c an cient re raction , then C be foun d in are from the expression 10 C 10 e 1 1 7 in which is the distanc P sin 1 between the two stations and p is the

. i o f rad us the earth .

1 To Find the C oe icien t — co 7 . fi The efficient of refraction may be found in

f f i r o . : rec o either two ways , viz I , rom p e Or f Ob cal zenith distanc s ; , II , rom the a of s served zenith dist nces two station , the relative altitudes o f which have been - determined by the Spirit level . 2 B Reci rocal Zen ith Distances 7 . I , y p . — 9 if B In Fi . g , A and denote the posi

the e ts e s s th e ve In r por of progr of e U . S . Lak Sur y ’ and in Wright s Adjustm e nt of Obse rvations a defi ni tion is give n which m akes the coe fficie nt of refraction s e as the The fi i twice a larg above . de n tion as abow s m s be the o n e m e e e used ee to or fr qu ntly . 1 00

of i s e tions the two stat on , the angl s which the observers attempt to measure PAB A n of and QB . O account the

FIG . 9 . f of of raction the atmosphere , the a ray o f light from B to A will not be a straight line , but some curve more or less

1 02

r f to efraction which is ound exist , even at

h e s few t e same mom nt , at two station a

‘ miles apart . The observations should be simultaneous or nearly so . Under these conditions the coefficients at the two ends

f e i . e . re o a line becom equal ; , in the p ’ ‘ = ceding formulae m m . The error in this assumption will be greater as the distance between the stations is greater ; it also increases very rapidly with the dif ference in elevation . Making this substitution (19) becomes Z+ ZI 180 m (20) C

3 B zenith distan ces the di 7 . er II , y , fi ence of level being known — There are two cases : (1) single zenith distanc es and (2) reciprocal zenith distances . (1) Measure the zenith distance o f the s f ff tation , and rom the known di erence o f level compute the true zenith distance ; the difference between the true and c omputed z enith distances is the refrac

e tion angl , which , divided by the sub h e e t e f . t nded angl , is coe ficient sought

'

. S. C . . Re 1 88 2 . 1 8 2 . U G port , . p 1 03

F Fi ; 9 rom g , CB AC h’ h ’ GB AC 2p+ h + h AB BA tan . %(C C ) CAB CBA tan . ( )

° z — CAB 180 mc,

c 0 CBA z + m .

S n ubstituting and reduci g, h’— h — m c ot 2 mC 7 0 o f ( + 5 ) tan . t 2p+ h + h D G eveloping tan . 5 by x = tan . x x + etc . 3

G= and substituting ij, we get

’— = — — 1 ~ Z h h k cot . ( 4 mC 7 0) h’+ h k2 1 + 2p

h h ' — = 1 Z+ mC "w oot g h’ -h k2

f m be f n rom which can ou d . (2) Also CBA= 180° sub stituting this value in remembering 1 04

= Z that and Z+ mC 0 and solving as before we get h’— h Z " 1 0) tan . k h' — h kz

0 T Z “ “ 0) 90 + ic o Z — Z= F Z’ O and O F F' and finally m and m U C

truve Bauerfi end S ,T , and others have deduced rational formulas for computing the coefficient o f refraction from the o b served temperature and barometric pres

f of e sure , but such ormulas are littl utility , owing to the difficulty in the way o f get ting the temperature and pressure o f the atmosphere . 4 Laws o Re raction Ex . ri § 7 f f . pe ence has proved that the refraction is greate r and more variable at sunrise than at any other hour of the day ; that it

r e g adually diminish s in both respects ,

. C . . Re 1 88 2 . 1 83 . U . S G port , , p ‘ B s O e u ve . 2 1 4 . 1 riti h rdnanc S r y, p

1 06

in creas temperature , but increases with ing atmospheric pressure ; in general its value depends upo n the law of the dis tri ” bution of e t mperature with the height . There is an irregular effect o f refrac “ ” tion , usually termed boiling , due to i s of i vary ng den ity the atmosphere , wh ch causes the target to vibrate rapidly through a small angle rarely excee din g but whose effect cannot be calcu lated . For a table showing the diurnal varia f f G . Re o se . . tions re raction e U S . C —2 1 . . . 1876 . 36 . G port , , pp ; also U S C 1 2 5 1 R 883 . 9 3 0. eport , , pp and

V alue o the C oe icicnt 75 . f fl

De Lam re rom o s erva io ns in b , f b t — Fran ce 1 ’ Be a s Trait" de Geo es ie in g t d , ran e fo r Summ er F c , ’ B at s T ait" de Geodesie in eg r , ran ce fo r Win te r F , ’ a i e Geo es ie in Beg t s Tra t" d d ,

fi an e fo r m ean c ,

C . 8 . Re . U . S . port 1 07

Bess e ro m O e ra io n s in Prus3 1a — l, f p t 3 u u a _ 3

v s a — 3 S ru e Ru si . t , o ra f C beu ,

‘ Mean from all o bs ervations in Fran c e Chauvan e t gives Th Br 00 e i i r r 6 0. s O u e 0. 0 35 S v . . 7 8 t h d . y i r ra a Do fo n o t o in he s e . . ys c r ss g t cro ssin g w E n lan n ar h S. in Ne e t e S. U . C . g d — 5

in New E n an fo r S . S. U . . C gl d — sm all elevati on s 5 in New E n an e w en U S. . S. e . C gl d b t rim ar s ation s — 5 p y t . f o r — in in e io r o c un t . U S. . S. t 7 . C r y Lake urve in en ra Illin o is S. S U. y C t l

’ 1 Chauvenet s c s m . 1 . 1 77 . . Pra tical A trono y , Vol , p ’ e s M em l i . N Davi es and P ck ath atica D ctionary. p . 20 O O d e u ve . 5 12 . O r nanc S r y , p h r v Re . 428 . . . e u e U S Lak S r y port , p ‘

C . R . 1 8 6 . O U . S . 8 fi ’ O Cl e s e e s . 283 . ark G od y , p

fl . . C . . 1868 . 62 . U S S , p

A 2 ME L RT . . R ON O R C E E T IG T I AL V LIN G .

B bserved A 76 . O n le o Elevation y g f . D Fi 1 Let . 0 , g , represent the position o f E the observer, and the station whose 1 08

to termi elevation is be de ned . Let A the observed angle expressed in seconds = of arc ; k the distance between the sta tions .

iff of dh a f The d erence level, , is m de up o HF FE " to , due to curvature , and , due the n f n n of refrac a gle o elevatio . The a gle k tion EDB = mC = m The true p are 1

1 1 0

The last term is plus for angles of ele f f vation . The chie source o error in this as in all other formulas for trigonom et rical leveling 1s in m .

FI G . 1 1 . B 77 . y the observed zenith = of the sea Z the meas ured zenith distance h= the elevation , .

. Fi 1 1 sought Then in g - 1 1 1

1 — cos . C ° o+ h 2 2 2 0 sin . 0 . 0 p £ sin $ sin . C 2 c os . os 0 o . C c . 5 c s C tan p . %0 tan . C .

S G is a 0 ince always sm ll , assume tan . $

. hi s tan C , w ch , s ub tituted above , gives h 2 tan . C.

fi n d e Z mC — To C , notic that + + (90 C)

71 9 f C = there ore which , sub 11 72 st ituted in the above , gives Z— 90° h= 2 p %p tan . 1 _ m m 2 Z tan . ( (30)

7 B the zen ith dis tanc e obs erve a 8 . y d t — = one s tation Let dh the differe nce in level between the two stations . The tri

s m ABD .

ABD= Z+ mC — C BAD= 90° — (Z

S. C . . Re 1868 . 1 27 . U . S port , . p 1 1 2

Z — . ( + mC %C) dh= kcos Z m — sin . ( + C C)

B to 79. y ’ eliminate refraction — Let Z and Z be

C . . Re 1 868 . 1 26 . U . S . S port , . p

1 1 4 should be measured and made a part o f the record . The correction to the z enith distance fo r the difference in height of the target and telescope above the ground may be' computed by the formula : correction in

= — j — s . d if econds 7 , in which is the d m 1 k s . ference in height and k is the distance between stations .

2 Limits o Prec s — § 8 . f i ion Reciprocal z enith distances measured at any two sta

e o f tions at the same mom nt time , or under the same supposed condition of a s We tmosphere , give the be t results h n o b reciprocal , but not simultaneous , the s ervations should be made on different

a -o f days , as in the c se horizontal angles , in order to obtain as far as possible a mean value of the difference between the respective angles and an average value o f f the re raction . The same care should be taken when -the zenith distance is meas ur ed at one station only . The condition of the atmosphere and the relative refraction may be so different 1 1 5 at stations situated more than twenty i r n e m les apa t , that , as a ge eral rul , the difference o f level determined even by re c ipro c al observations cannot be relied upon fo r the desired degree of accuracy at distances greater than about twenty

e of miles , unless a very large numb r

e e measur ments hav been made , under

e N e the most favorable circumstanc s . otic that the differenc e o f height determined by trigonometrical leveling depends upon the co efficient multiplied by the square o f the s f e e di tance , and that , there or , ther is a lim it to the distance fo r which any as sumed mean coefficient can be depended

fo r l hi upon accurate resu ts . The gher the the e e re elevations , more r liabl the l s u ts . f 83 . e o . § In the final r port the U S .

L e 544 t h ake Surv y (p . ) it is sta ed t at the probable error o f determining a difference o f level in E aste rn Illinois by reciprocal e e 8 to 10 z nith distanc s , using separate as 2 6 me urements , made on to days , over li 16 to 20 i e nes m l s long , was somewhat less than 1 foot . 1 1 6

1 C . . . S. G R 6 . . 87 The U S eport, , p 345 ns of tri onom et , contai an account g ric al e e C f ni i 14 l v ling in ali or a , over a l ne

e mil s long , in which the mean probable

o f e i 1 1 error r ciprocal zenith d stances ,

e 5 hourly obs rvations on successive days , was meter in 600 meters ; fo r single

zenith distances the error was meters . The same volume contains an account o f G in angular leveling in eorgia , which 379 (p . ) it is stated that the probable er of 20 i 4 error p line about miles s . 87 meter or Wof t he horizontal dis

tance . Probably the above may be cons idered

of e as examples the best work possibl .

1 1 8

E is or nglish instrument a type . The third includes all instruments whose errors o f adjustment can be wholly eliminated

o f s Th by a system double observation . e levels employed in geodesic leveling are of this class and are generally known as

e of lev ls precision . There is considerable variety in the f of i o f nl wo orm th s class levels , but o y t have been used to any cons iderable ex tent in this coun try— the Swiss or Kern L S M i level , by the ake urvey and ississipp R C mi of iver om ssion , and a modification n S f e C t the Vien a or tamp er l vel , by the oas ns and Geodetic Survey . The co truction o f the the two are similar, hence only e f r latter will be described h re . The orme of is described in the final report the U . 5 L S . 97 S . ake urvey , p , and described ’ and illus trated in Jordon s V erm es sun gs I 4 40 . kunde , vol . . , p .

U oas t Surve Level— . S. C § 86 . y The Fi Th instrument is shown in g. e telescope may be reversed end fo r end and revolved about its Optical axis the

R 1 79 . 20 2 . m C . . e . 8 . Fro U . S . S p ort p 1 1 9 two positions in which the horizontal fi ni e thread is horizontal , being de t ly fixed s by projecting pin . The level can also be reversed end for end independent of ne of the telescope . O end the telescope and level can be raised and lowered by

N the m ic rom the micrometer screw . ear - eter is a cam hook , by which the weight o f the superstructure can be raised o ff u s the micrometer d ring tran portation . Under the telescope are two false wyes on lever arms by which the telescope can be raised out of the wyes fo r transporta s m r tion . The whole in tru ent is secu ed to the tripod head by a brass plate which f of e fits over the eet the l veling screws .

r f e 4 The apertu e o the t lescope is 3 mm . 1 f 41 0 ( % inches nearly) , ocal length mm . 16 f ( 5 inches nearly) , magni ying power f i 37. The value o one m llimeter o f the = ” al 1 . 5 1 5 level sc e is ( inch 37 . or = f di radius 800 eet) . The aphragm is glas s and has two horizontal and one ver

e tical line rul d upon it . The two hori z o ntal lines used as stadia hairs to , are o f determine the length sight . 1 20

Asmaller size of this instrument is also

. of in used The weight the larger one , lus iv o f 4 l c e 5 bs . tripod , is ; the smaller 2 weighs 3 lbs . The d — h § 87 . Ro T e rod used with this instrument was a target rod made of 3” ff f pine , by sti ened by a strip ast mi of f ened along the ddle each ace . One edge of the main strip has a self-reading A graduation to cm . upon it . brass strip , i h . s t e of graduated to cm , let into side the strip ; it extends the whole length of ‘ fa the rod , being stened immovably at the middle . Its temperature is deter mined by a thermometer let into the side f o . the wood The target , which is moved by an endless chain , carries a short ivory hi i s mm . cale , graduated to , w ch sl des over the brass strip ; the rod therefore m li f f reads to il meters . The oot o the rod is a rounded piece of brass resting in a correspondin g depression in the foot * A an d e p1ate . handle a disk level enabl the rodman to keep the rod vertical .

Would it not be bette r if the e n d of the rod were hollo wed out to fi t a projection on the foot plate "

1 22

— f But the back and oresights . to employ c heck o f double leveling (as will be de s fo r cribed presently) , a correction inequal ity must be applied to the rod reading . The inequality o f the collars can be de termined by observations with a striding

the o f level , exactly as pivots an astro n om ical transit are examined ; co ns e quently it is unnecessary to describe the ’ method here (see Chaven et s Practical

1 . . . 53 S. Astronomy , vol II . , p . , or U C

1 . orre . R G . S 880 c c eport , , p The tion to be applied to the rod reading at the d distance D will be sin . in which d is the pivot correction in seconds of

arc .

90. e The absolute valu , and the uni

f o f o f ormity the graduations the rod , the f of s the o f c oe ficient expan ion , accuracy

s e the thermometer, and the di k l vel s hould be tested carefully The observer should also know his ordinary inaccuracy in performing the different parts o f an

o bservation .

orrect or 91 . C ion f Curvature and Re

n C urv r — fractio . atu e To compute the 1 23

fo r r AD Fi correction curvatu e , let , g. 14 of , represent the line apparent level, and AB DB the true level . is the cor for r B G recti on cu vatur e . y eometry AD2 = DB 2BC DB N DB ( + ) . eglecting ,

FI 1 G . 4 . as it is very small in comparison with 2BC s the of , and repre enting length s k of e ight by , and the radius curvatur n for u e by p, the correction c rvatur k2 BD 1 24

Fo r BD f i in eet, and the d stance in m iles thus becomes BD kz .

’ r — D Re action . of f is . the true position D h the target and the apparent . It as 1 . C . been shown (Art , hap IV) that the ’ of f DAD to mC angle re raction is equal , in which m is the coefficient of refraction d A D an C the angle C . h = ’ _ DAD k tan . 1 o . , sin . 2 mic DD' = k DAD’= kDAD’ tan . tan . 1 P

It will be impossible to select a value of ‘ m which shall be true fo r all cas es ; but s ince the line of sight is always near the n ground , and since the observatio s are generally made in the morning and eve n s ing , when the seeing is be t , a large value should be chosen . The Coast Sur = 0 . vey uses m o. 07 Total C orrection — The correction for c urvature and refraction to be applied to

. . C . 8 . R e 1 88 2 . 1 77 . U S port , , p

1 26

A A . f ment ; ,, 2 , etc , successive positions o B B o ne etc . rod ; and 1 , 2 , , successive posi o f tions the second rod .

o f i I is the method ordinary level ng, and may be called single kveling with o ne d ro .

0 sin le levelin with two rods B II is g g . y of t the use the two rods , less time in er venes between the backsight and fore it ef e e as sight ; is ther or more accurat , there is not so much liability of change f f in the plane o the line o sight . It is also s more rapid than with a single rod . Thi L is the method used on the U . S . ake * M R Survey , and the ississippi iver Sur veys . T double levelin with one rod III , g affords a perfect check against errors of dif adjustment and observation , since the ference in reading of the two foresights should be the same as the difference of f the two backsights ollowing .

double levelin with o rods IV, g tw f combines all the advantages o II and III .

’ E i s C e ee Re . . A 42 . 1 880 . 2 7 . hi f ng n r port , U S , , p “ Miss ss R ve Co mm s s 1 Re 188 . 1 i ippi i r i ion port , 1 27

e It is the method us d on the U . S . C . * um e s G . Survey . The n ral adjacent to the rods show the order in which they are sighted upon . du licate levelin V, p g, is another method

e s . be sometim s u ed Two rods are used , f ing placed as in the figure . A ter having

A B the observed upon I and 1 instrument is pulled up and reset a little to one side and the two rods sighted upon again . This method duplicates the work as far m as instru ental errors are concerned , but is not an independent check . The Obse vation — f e § 93 . r A t r having planted the tripod firmly and leveled the i r of nst ument, read both ends the bubble , estimatin g the fraction of a division ; next read the position of the two (or three) wires on the rod ; then read the bubble again for a check to eliminate any R fo r change . everse the level end end , ° and turn the teles cope 180 about its O t e p ical axis , and r peat the operations as above . The first reversal eliminates any inequality in the lengths of legs of

* U . s . C . G . 8 . Re 1 79 . 20 port . 8 , p 6 . 1 28 the stridin g level ; the second eliminates of li any error col mation . The mean of the several readings must be corrected fo r

ff of e the di erence in position the bubbl , for of and inequality the collars . Instead o f reversi ng the level and i telescope at the same t me , the observa tions are sometimes made as follows

R ro d e the ead upon the , r verse level , and e read again ; . reverse the t lescope and

read a third time , then reverse the level r and make a fourth reading . The fi st method seems the better . If l 94 . there is a mil ed head screw of under one end the telescope , the bub ble can easily be brought to the middle i if each t me ; there is not , it is better to bring it nearly to the middle and apply a nl correction . It is not enough to read o y to one end , since the bubble is liable change its length with a change of posi

tion or temperature . On 95 . the coast survey the method of observing differs slightly from that * E of described above . rrors level and

. C . 6: . . Re 1879 . 206 U . S G S port , , p .

1 30 collimation are eliminated by reversing bubble and telescope on each backsight and foresight ; but each observation is of i a single w re on a target . The target is for ff set but once each station , the di eren tial quantities being read by the micro m e of et r under the eye end the telescope . This seems not to be as good a method “ as the above ; there are two objections aside from the time and labor required to t F s f se . u fi the target irst , there is no cient check agains t errors in reading the h Of . t e positions the target Second , micrometer is read fo r a central position of the bubble , the telescope is then moved to bisect the target and the screw read f again , there ore there is no check on the ” stability of the instrument . L i ht — n n th o S O . . § 96 . e g f g the U S f . G. o e C Survey , the length sight rang s f 50 to 150 rom meters , according to con of dition ground and weather, the aver 1 10 dista ce be age being meters , the p tween the two rods o n the same side of On the the instrum ent being 20 meters . L S * 100 ake urvey , the maximum was

Re . 598 . Final port , p 1 31

t P L me ers ; on the russian and Survey, 1 879 50 since , it has not exceeded meters , fo r * except river crossings . The attempt is always made to place the instrument half way between the two stations ; the rodman approximates the i distance by stepp ng , and the instrument On man measures it by the stadia hairs . L ff the ake Survey , the di erence between c orresponding back and foresights was n ot 10 allowed to exceed meters . — rces o E rror . P 97 . Sou f robably in no o ther kind of ins trumental engineering is it as important to distinguish between c s as ompen ating and cumulative errors , Fo r in leveling . convenience in discuss f of ing them , we shall classi y errors lev f : 1 E eling as ollows , Instrumental rrors ; 2 o d E 3 E of O , R rrors ; , rrors bservation ; 4 h Pe E and t . , rsonal rrors

1 Ins trumen al rrors — . t E The princi pal instrumental error is due to the line o f l sight not being paral el to the level , which may be caused by impe rfect ad of justment , or unequal size rings , or

’ i s s m e O se v ns . 375 . Wr ght Adju t nt of b r atio . p 1 32

If is bo th . the telescope slide not

' will also straight, or does not fit snug , it Of u ins tru produce an error . co rse the ment must be focused So as to eliminate parallax All of these errors are c limi nated , whatever their value , by set ting the instrum ent midway between the turning points . It has been foun d that appreciable er ro rs are caused by the settlin g of the in of the strument on its vertical axis , and settling of the tripod legs into the ground ; in spongy or clayey ground the tripod legs are sometimes grad uall f h y li ted up . These errors , thoug im small in themselves , are more

portant than is generally supposed , inasmuch as they are cumulative ; but probably they would be appreciable in n only precise leveli g . They can be eliminated by running the line in the op osite di p rection . A small source of error arises from the fact that the adhesion o f the liquid to the sides of the glass tube prevents the bub ble from coming precisely to its true

1 34 inated by noticing the position o f the f bubble a ter setting the target , and can be still further reduced by shading the instrum ent . The Indian Geodetic Sur vey proved conclusively that the error was appreciable even when the ins tru * ment was shaded .

2 Rod E rrors — r . The principal rod e ror is in not holding the rod vertical ; this may be remedied by attaching a level or by waving the rod . In waving the rod , care must be taken that the front face is not lifted by resting upon the back edge

the is when rod revolved backward . With telescoping target rods when ex of tended , the slipping the upper piece , after the target has been pronounced cor f i a rect and be ore the vern er has been re d , is a source o f error . The target itself be may slip , but this is not so probable ,

e cause of its less w ight . An other source of error is the settling o f the turning point , due , in coarse or sandy its to of soil, to own weight , or the impact setting the rod upon it . The resulting

’ u v ce . 18 1 . Jacks on s Aid to S r ey Practi . p 1 35

error is cumulative . The remedy in the

first case is to use a long peg , or to rest the rod upon -a triangular plate with the corners turned down slightly, or with f Spikes on the under side . This oot

o n plate with a convex button attached , is a which to place the rod , better than for peg all cases . Whatever the turning n poi t , the rod should never be dropped upon it . Fin h ally , another small rod error is t e error in the graduated length . This affects only the total difference of eleva

e tion betwe n the two points . This is a o f much more important . source error with the numerous home-made self-read ing rods now in use , than with the rods

e I mad by regular nstrument makers . “ An important source o f error in spirit m leveling , and one very co monly over

in o f looked , is the change the length the leveling rod from variations of tempera is ture . It quite possible that errors from this source may largely exceed the ” errors ari sing from the leveling itself.

’ A m O v s 2 . Wright s djus t e nt s of bse r ation , p . 37 1 36

Err s o Observations — 3 . or f The prin cipal error of observation is in reading the position of the bubble ; even if the i bubble is kept in the m ddle , it is never Ev r theless read . e v leveler should kn ow the error on the rod corresponding to a given distance o f reading of the bubble ; he then knows how accurate ly he must .read the bubble fo r a given de of gree accuracy in the results . Another source o f error is the moving o f f e f the bubble a t r being read , and be ore the sighting has been made . This move ment of the bubble may be caused by its

f has being read be ore it come to rest , by disturbing the instrum ent by stepping in near the tripod legs , by turning the strument slightly in azimuth , or by raising or lowering one end o f the telescope in f o f ocusing , or by the action the sun or re— wind . The bubble should be read f a ter the target is nearly adjusted , or f- f with a sel reading rod , a ter the reading has been m ade and before the rodman is signaled to move on . These two prob ably constitute the chief sources of error s in levelin g operation .

1 38 each hair and the mean used as the rod

Fo r e reading . a singl observation , a tar get rod is more accurate than a self-read ing one ; but three observations as above are probably more accurate than a single

be observation upon a target , and can made in about the same time . When very long sights are required to

w e e o f be taken ith the l v l , another source r h s e . t e erro must be con id red , viz , curva f f w ture o the earth and re raction . O ing 220 f dis to these two causes , a point eet f i tant appears about t. too high ; th s error increases as the square of the dis if h tance . It is wholly eliminated t e in strument is always exactly half-way be f tween the turning points . Re raction is not always the same ; its mean effect was used in finding the above correction .

if r Consequently , there is abnormal e fraction or a change o f refraction be

e o r . twe n sights , a tremulousness or “ ” o f e boiling the air, small rrors may Th result . e only remedy is to shorten

o f fo r the length sight , or wait better atmospheric conditions . The atmosphere is 1 39 usually in the best condition for seeing just before sunrise and a little while before sun s et f , although the re raction is then greater . A cloudy day is better than a clear one . There is a possibility o f error in record ing the observations and making the c omputations , but in precise leveling there are so many checks that there is no

fo r e ,probability s rious error in this respect . — P rs h " 4 . e onal E rrors T e errors pre vio usly described are liable to occur with any observer ; they are due chiefly to the

u e t e o f instr m n s and to the natur the work , and would probably not materially differ

r le . We fo equally skil d observers come now to a class of errors which depend mainly upon inaccuracies peculiar to the individual . W of f ith a target rod , errors one oot - and one tenth are not uncommon . The only check is fo r the rodman and observer to read it independently and compare notes ; but this is inconvenient and not W f- always possible . ith a sel reading rod to this error is less liable occur , especially if three hairs are read . 1 40

F di inally , each in vidual has errors i s f peculiar to h m el , or to the work he is n e e doing . O may r ad a target higher or lower than another of equal skill ; or one observer in reading the position of the bubble may have peculiar views as to

e of e what constitut s the end the bubbl , or he may habitually read the bubble so as to get a distorted view of it through the glass tube ; errors from these causes are compensating . Again , the target may be better illum in ated on foresights than backsights , as in working toward or from the sun ; in this case the error will be cumulative . W f ith skill ul observers , all such errors are quite small and generally cancel them f selves . In act , the errors here classed as personal are possible rather than demonstrated as actually occurring ; and i yet , there is noth ng more certain than o f s that in any series accurate observation , there is a difference between the results ff hi ff of di erent individuals . T s di erence is known as personal equation . In long n of r li es accu ate leveling , it has been

1 42

as is tion , done in India . It may be still further increased by reading the back sight first at each alternate time the in strument is set up . The effect o f this class of errors may be eliminated by each observer duplicating this work in the opposite direction un der as nearly the same conditio ns as possible .

Limits o Precision — § 99 . f The prob able error per unit o f distance is gener ally adopte d as a convenient measur e of i to h the precision reached . Accord ng t e of f theory probabilities , the final error o

o f r afi ected l a series obse vations , on y e l as th by accid ntal errors , wil vary e square root of the num ber o f observa tions ; hence the error of leveling a num ber o f units o f distance is assumed to

e of i vary as the squar root the d stance . This as sumption would be true if acci nl dental errors were the o y ones made , and if the number o f observations were s trictly proportional to the distance lev if f l . e e o e ed i . e , , the l ngth sight wer constant ; but in the preceding article it 1 43 was shown that leveling is afi ected by an error which is nearly proportional to the f distance . It has requently been noticed

s o f that , con idered individually , the errors a number o f short lines were well within the limits which were prescribed to vary

o f et as the square the distance , y when the sum fo r several lines were considered the total discrepancy would exceed the fo r limit . In other words , a line leveled in only one direction the error is not s trictly proportional to the square root of the distance On e part of the error is proportional to the square root o f the distance and anotheHr portion varies nearly as the distance . ence the shorter the di stance , the easier to attain a limit pre s cribed to vary as the square root of the di stance . “ According to the Geodetic As socia t of E e of ion urope , lev ls precision exe o of E uted late years in urope , show that the probable error of a line of levels o f precision should never exceed

f t . 1 44

is 2 tolerable , dist . in km . is a fair

” precision . The Coast Survey requires

Of late years the Coast Survey’s and M R S ’ ississippi iver , urvey s work are con siderable within the limit of

2 Mi s kilom . The ssi sippi River Commission’s limit is “ "/ k f V mil 5 . t es ilom . ) . The limit on the British Ordnance Survey is f ” t . per mile . Results o f levelin g are often given of apparently greater accuracy than the limit abo ve ; but regulari ty of result and even ness of error is of more importance than is occasionally small disagreement . It usually the latter that is recorded . If the error was determined by dupli eatin di g the work in the same rection , and if ‘ especially at the same time , as by ff methods III , IV, or V, the di erence will

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