THE GIFT OF .S)^jT<-^Lt^or arrange- ■ ments made for their return during borrow- er's absenc^, if wauteU; Books needed by more than one person are held on the reserve list. ■ . . Bbokss / of special - value and ^ft books, when the giver wishes it, are not allowed to circulate, Cornell University Library QA 275.C89 Notes upon least squares and geodesy :pr 3 1924 001 898 745 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001898745 NOTES LEAST SQUARES AND GEODESY PREPARED FOR USE IN CORNELL UNIVERSITY CHARLES h^ CRANDALI. Professor of Railroad Engineering and Geodesy ITHACA, N. Y. ANDRUS & CHURCH 1902 3> ■A. 1 17475 Copyright, 1902, By Chari,es L. Crandai,!,. C H T S 8 T S-. iii PART I. LSAST. SOJASSS'. Chapter I. Meitiod of Least Sqaares» 1. Introdocfti'on. 2. Meair-S'guare Error. 3. .Law of Propagation of Error. 4. Lair: of Propagation of Error. 5. Tlie Simple Arithmetio Mean. 6. The ITeighted Arithmetic Mean. 7. Controls. 8. Cldsenesa of Compitation. 9. Independent Observations upon Independent Qiantities. 10. Control, (Tormal Equations. 11. M.SiB'S' of the Dntoowns. 12. M.SvE. of an Observa- tion. 13. Solution of Hormal Bqaations. 14. Form for Solution. 15. Independent Observations upon Dependent Quantities. 16. Control. 17. Example. 18. M.S'.E.of a B\>notion of the Required Qiantities. pp.1- 16. Chapter II. Theory. 19. Principles of Probability. 20. Probability Carve. 81. Form of f(^J, 22. Codstant C. 23. Value of Probability Integral by Series. 24. Degree of Precision. 25. Constant £ . 26. Average Error. 27. Probable Error. 28. Graphic Representation. 29. Principle of Least Squares. 30. Rela- tion betwgen Average, Mean Sqaare.and Probable Errors. 31. Limit of AoaiP' acy. 32.Rejection of Do!})tful Observations. pp. 16-23. Chapter III. Application to Triangulation. 33. Triangulation. 34.. Station Adjustment. 35. Weighting. 36. Figure Adjustment. 37. Adjustment of Quadrilateral. , 38. Hamber antl'-^ParBatioB of the Side Equations. 39. Adjustment of Secondary to Primary Work. 40. i4.S.E. of any Side. 41. Approximate Adjustment for Azimuth. 42. Approx- imate Adjustment Betifeen Eases-. 43. Adjustment for Latitude and Longi- tude. 44; Trigonometric Leveling. 45. Adjustment of a Compass Survey. 46. Adjustment of a Transit Survey. PP. £4-33. PART II. GBODESr . Chapter I . Introduction . 1. Geodetic Survey. 2. Historic Outline'. 3. Historic Outline. 4. Geo- detic Work in )ihe United States. PP' 1-4 Chapter II. Triangulation, Reconnoissanoe, Signals. 5. Primary, Secondary, Tertiary, Triangulation. 6. Triangulation Systems. 7. Elevation of Signal. 8. Hints' in Selecting Stations. 9. Base Lines. 10. Reconnoissanoe, Primary Triangiiation. 11. Secondary and Tertiary Triangulation. 12. N.. Point Problem. 13. Tiro-Point Problem. 14. Di- rection .of .Invisible Stations-. 15. Oatfit. 16. Signals. 17. Pole Signals-. 18. ■Diameter and Height. 19. Signals ifithout Phase. Z). Ele- vated Signals and Observing Stands. 21. Heliotropes. 22. Sight Signals-. 23. Station Reference. tt.s-19. Chapter III . Instruments And Observing^ 24. Development of Angle Instruaenls. 25'. [formal 7d.saon. 26. The As- tronomical Telescope. 27. Ifeignifylng Poirer. 28. Measurement of Magni- fying Pouer. 29. Intensity and Brightness. 30. Field of Vie». 31.. S'pherioal and Chromatic Aberration. 32, Eyepieces. 33. Cposs' Hairs.' 34. Tests- of Telescopes 35. Level Tubes-. 36. Qradoated Circles. 37. Mi- crometer Microscopes. 38. The Ran of the Micrometer. 39. Errors of Graduated Circles-. 40. Repeating and Direction Instrojients. 41. Ad - justments. 42. Determination of Instranental Constants-. 43. The MetboQ ojf Direction Observations in Horizontal Atlgles. 44. The Method of Sim- ple Angle Measurement. 45. The Method of Repetitions-. 46. Conditions Favorable for Observing. 47. Accuracy of Results. 48. Forms for Record. 49. Phase. 50. Eccentricity. PP. 30-39 W. C » T S If ? S; . Chsp-ssr IV. Base LineS) Sit- Bas© tiae Sites. 52. Early Pgrms of Base Apparatas. 53. Bache • ffll]?demsB Apparaios./ 53*- Porrs Apparaias. 5S. U.S-,L»S'. Bepsold Appa- ratus. 56. ibaaSa' Apparatus. 57. 0:.SvO.S'.Seoondary Apparatus. 38. U.S. C.S. GridirOB''CoT«pensating ApparatiiS'. 59. O.S'.C.S.Dirplex Apparatas. 60. Standards' of Dea-gth. 81. Comparators-. 6S. Meroarial Thermometers. SS'. Length of^lpparatos. 64. Defects and Difficulties. 65. Field Work. 66. Tape Ueasurenents. 67. Correction Pornral-as-. 38. Seduction to Sea Level. 39, 'Aooaraby of Results. pp. 40^50 • Chapter 7c. TrigOQometrio And Precise_ LevelinJ. "70. OTservationsw 71. Differeaoes' of Height from Observed 'Zenith Dis- taacesi 72. Coefficient of Ref?acUoD. 73. ObserirSd Angle of Sleya- tion' itt aecohdst 74. Reitootio.B for Difference in Height of Telescope and Object above Station Mark. 75. Zenith Distance qf Sea Horison. 76. Instrumsntsi. 77» Rods. 78. O.S. C.En;gi?S. Ketbod. 79. French Govt. MethodK 80. 03.3-.8.S. Method. 81. Cr.9»0.& G.3; M?thod. 82. Inequality of Pivots'. 83. Bod Correction'. 84. Accuracy and Cost of Results. 85. Datam. pp.51-60 Chapter VI. Tapographio and Hydrographie Surveyifi-g . 86. Topographic Surveying. 87. ffydroigraphlo Surveying. 88. Field CoomunicatiOBS. pp.&0-61 Chapter V"«o/nL = i-i« 3. LAff OP PBOPOSATIOli OF ERROR. Get -X = a jMji ^g Mg ± \ "a ^*^ irliere a, .a.^ ... a.„ , are coastaats uaaffected by error, aod M, ,M^ ,, li„ ,ape observed iodependent quantities witli the iii.s.e'ls.,tii^»'-"^'n ,are the errors for different ob served values of H. ,Vi„ U^ . the errors in the corresponding values of X irill be. S'gaaring each.lins and addiirg. (V5, Positive and negative errors of the same maenitude being emallv li- able to oooar. by axioB 2,%1, the products 4 2 i.J^agl^li5-i:2a^ aj.\'^^— iSag 'S^'^'-''"^ Trill tend to foot up zero {approaching it nearer the greater the nam - ber of observed valuesjand may be neglected. .'.dividing (b) by n.and remembering the definition of n.s.e., %2, (3) (4> In the general case, ujiere f ( ) denotes asy fanotibn. — If the different observed values be sirbstita«ed for the trne values of the observed quantities, we shall have fzpanding the secoad inembecs by- Taylor's theoren.aad sa-pposing the ob- servations accurate enough so that the squares products and higher poir* era of the A*^ i»ay be neglected]^ «-^*^.^^-^*|^-^ *^v)^. ^ - - - w-Kwd ^ =. i(f^ ,wVxc\v vJ 9. WLekii -p\o.t«. =.-2.. -1^^ Bu.\, S-U.V *ttt.>fc.tWlC^^ Similacly-TiE uay extend to 3 or Bore variables. as assumed above. m^hi^'^ ''' Bo. a.) ARITHMETIC KSAH Ihese correspond to (a), .• from (4), \dV Bx. 1. find the m.s.e. ia the length of a city blotk 500 ft. long meas- ared *ith a 130 ft. tape haring a m.s.e. ia its iength of.Ol ft. Ans.O. OS ft. Sx. 3 Find the m.g.e. in the length of a city block 500 fset long meas- ured ?iith 5 100-ft. tapes each ?Tith a m.s.e. of .01 ft. „ Ans. O.Oa ft. ex.3. In the triangle AEC, AC or b lOSO ft. irith £, - 0.1 ft.; A = 50° ifith t^=' 10" (in aro.=10 sin 1' );. B = e4' rith %.= 10". Find the m.s.e. for EC. or a. H) reaaces to a = b sin A = 903.44 ft. 3-1 n B sin A_ a df = dM, da db df ^ da dM^ dA sin B b b cos A .85 a cot A a.^758 Fi [7*J ilStyl + nS2 .(6)' can be irritten, (■:x„ - «.) + fx„ - l!_) .+ (x„ :«„)... =0 1 a « ■ or [vD = (8) Substituting and dividing by n. . s e= IV--] /n +S'- The most probable valae of 5'. the error of :x .is usually assumed to be the m.s.e. oif the mean itself.or Z^^yja. Substituting, f. = [7»y n + cVn t'-= [yi]/ (n-1) (9) From (7), Cj;^ [ySl /fc- June 21 (T 47'™ 41.154 .040 .0016 " 22 41.171 .057. .0030 23 41.133 .024 .0006 • 24 41.110 .CJ4 .0000 '29 40.995 ._119 .0142 mean. X- -»- ^ ' 47 41illi_^121 .123 .0194 t. =fM /(n(nTl)) = v/Olsi/ffl = 8P.031 9. fHE WBIfflTED ASITHBSTIC MEAN. An observation is said to have the ireight n.when its m.s.e. is equal to that of the mean of » observations of Height unity. If theat' is the m.s.e. of an observation of seight a- oity.and t, £...., are the m.s.e's.for reights 9, .w,..»e have from (7), or. V/»x '^i'c:- (11? i.e.. the Heights are inversely as the s quares of the m.s.e^. If the different values of a quantity, K^ ,lfg ,Mg ,..., have the ireights i±,^? ."3 each value being suppssd . to be the mean of b- values oi iflight unity, the sum of the original values can he found by multi - plying each mean by the number and adding; the average can then be found by dividing by the total number. I.e., the arithmetic mean, x„ =(,M^Tj + Mg Tg + Ug »3-)/ftrj+ '<-,+ V*' ^[il^M (12) The m.s.e. of the mean, £0 = ^/>^1 (13) (12) can be written, (x„- «,)«;, +(Ti,- M^)w^+ (x„- «,),w,a; ... =0, i.E.,i[«w] = (14) As_iD i S. „ «■«.._ if add 1^°° °'"^''^°° ^^ squared, then multiplied by its. corresponding if. and C'»'"l =[» »5t ^K' ^ ■- SM (a) Bq.lB.) CL0SSNSS3 OP COHEUTATION. 5 The obser7ations itlth npights " give errors d; the corresponding er- rors for ifBight anity ?iould most probably bea/ff", fpom tne relation (11) betireen w^igb^ts and m.s.e's, |[)icf]is.'.the sum of the squares of the errors for weights unity, - nC''bs 0^. By 55. 5'=Co, -■C/.^ V,, (13^ Substltiiting In (a), nf*= Cu 7'-] +C'- <'^\r, 7»-3/(n-l) (15) Ex. 1. We folloirlng valued are giyen m Pri.Tri. Q. S.Lake Survey. p. 395, for the observed difference in longitude between Detroit and Cam t bridge. i^o.-^ 13 i J'UTlt 4 11 Mean=0 (J* 47^" 4.1? 163 0.5 -.117 -.059 .00634 4g.95§ 0.5 + .030 +.040 .00320 41.03S 1.0 + .003 +.003 .00005 41.030 1.0 + .016 +.016 .00026 41.064 1,0 - .033 -.Ora .00144 41-012 1.0 + .034 +.034 .00116 47 41.045 Sums +.001 .01296 t,= \/. 01296/ 25 = 8-:023. 7. CONTROtiS. Simple Arithmetic Mean. Since, 7, X,- «,, 7, X.- M^. 7, = :x,-M, . . .and n ,x t7'D= nX- 2 x.M + [«^ Or, M = [«9-L»0V n \ (17) Also. from (8), W = J ^ "' Weighted Arithmetic Mean. Since 7, = X.- K,, 7^ = :x„- K^, 7, = x.-Mj', and by UZ) . C«7»J =:x;[w] 2;x^f mJ + (» M^J =M. (13) Also. from (14). L7 wD =0 J It may be noted that .the left hand places as far as they agree may be left off from the yalaes of M.or any constand s.ubtracted,iTtiene7er it irill, simplify the numerical computation for (17) or (18). In £z. 1,§S. ire-haye for the different 7alues of M .subtracting 41 from each; ,154. .171,. 138.. 110, -.005. ^ijuaring and adding, .0841 Adding andTijlatfing, LiflV a = .0645 .019&' nearly checking [,7^]. The mean x, wiien multiplied by 5 is +.002 greater than [jQsa that Ij] should f+.002 instead- of 0. '" Ex.2.. In Ex. '1,66. subtracting 40 front each H, ly. M'] = S. 48041 y>^^[/b- =(tk.4.*-')VC''v-*-«->''+ - - -V"- Bence If le alloii- the difference between t,,tandf;^to be O.OIC^, i.e., allow the D>s.e. to be increased 1% by inaccuracy in compftation, which would ap- 6 LEAST SGDARBS. («9.Pl^.l, pear safe. then .01 = oV2£; ,or c 14£^ ( 19> or. the error of ooaiptitation caa be 1416 of the a.s.e. irLthoot sensibly in^ creasing the inaccuracy of the result. . 8X.1. In a l-place log table the error ia the last place ^lUv-ary from to .5 .all values within these limits occurring mth equal fieoaeBCi a. The m.s.e: for this method of distribution o* "ror is aA/T ."here .a =«.«- greatest error. This ooold give.m.s.e. = -5 /V^ = >29 in tne 7th. place. An interpolated value.expressed as Mj +(Mg- »j1m, where Mj and Mg are the ad.iacent tabular quantities and m the percentage interval b^ween the corresponding numbers.irOBld have the follonlng ui.s.e's.for differ- ent values of m.the 7th place only being retained in the interpolation (Annals of Mathematics, II, pp.54-59:or Geographical Tables, p. Ixxxvi). y.s.e. 1/2 ,41 1/2 .35 The average u.s.e. 3 m.s.e. of .3 second 1/4 Ml ■ 37 1/6 177 ,33 179 .39 179 ITio" ■ 39 nill. tins be well i7ithin.-0»4. .In fieadelic iioti_ Is about the mininnn value "for Eorlzontal angles. A triangulation will be most exact. or the test most severe, when the an= gles of each triangle = 60". The change in log sin 60° for a ohangB of 1" Is 12.2 in the 7th. place so tiat the m.s.e due to inaccuracy of measurement .3 »C12. 2 = 3^7; i.e., c/t^ - .4/3.7 = l-l%,instead of the 14!6 allo^ved by (19). Again a m.s.e. of 1 : 1.000,000 is excellent base line jiork. The log of 1.000,000 is changed 4.3 in the 7th. place by a change of unity in the number so that c/£.,^= .4/ 4.3 =9%,instead of the 14 allowed by (19). J-plafiS lags are thus ample, for the best geodetic work. $- place logs are ample for, ( 19), in angle, in 1,000,000 in distance, t.^=.4/.14 -• t.9-, V9/I.22 »-9A .43 2.4' 7 or for the best city work. 5- place logs are ample for 24° 7 in angle, in 100,000 in distance or for the best railroad, or ordinary first-class field sprk. 4- place logs are ample for, 240", or say 4'. in angle, 7 in 10,000 in distance, or for the best chain and compass work.and much of the stadia work. S'ith suitable tables, like 7.ega,7-place; Bremiker g-place; Gauss' 5-plaoe; Encke says the times required for the same computation are as 3,2,1, re- spectively. Be also says, 9 places are sufficient for miautes and 1:4000 in sides; 5 places for S', and 1 : 40.000; 9 places for 1/Z'; and 7 places for 1/20", lijiits not as conservative as the above. 9. INDEPBSDEFT OESER7ATI0HS DKIB INDBPE8DENT .aJANTITIBS.,' In the general case of indirect observations let the equations be of the' form f'.(.X,Y,Z ....) , Mj = Might Wj .) 7 Mg =0 reight;rg (>S0) , M^ ,...,is greater f (.x,y,z in irhich the number n of the observed quantities M, than m,that of tne required ones, Xif Z The observations being imperfect, no set of values can he found for the unknojTOs which will not leave residuals, so that (20) woald be more cor- rectly written, f ( X-, ?, Z ) M, =7... ^ ■*' {21} f( 2,i,2 ) - Mg = vg wtiioh are sometimes called error or residual eauations He first find approirimatB values, by partial solution or otherwise for ■^■^''^ ^° that !(=,X^*:x. If-it^+y (x.y being so' small that terms containing the squares, products and higher mwer.? ma,, be aegleotea. without sensible error), tSen expand by Taylor 'rthelreSf as in § 3; ( 21) thus becomes + b 1^ ^ u.z + +1 3 ''1\ ^•^•) C0NT90L,S0RKAL SOJAHOSS. agx + bjy + ojz + . . . . tip , (22) a^x +bc>7 + ojz + .... +12 V2 y + Cgz + . 4I3 =73] «here a df/dX.. b df/d!f. , c = df/dZ. coii3t3at3 1 f(X^ .Y^....) -M. The most probable yalues for the corrections, x,r, 2 (it vill be proved later) «rill be those irhich Till make (V v*] = minimani. 'Hence since x.y-, z,..., are independent, d[ii v^/dx 0, d[ir y'J/d y = 0, d[ff y'^Vd z = 0,. or,ffj7j d 7/d X + WgTg d7.^ /dx + = 0^ wi/j^ d 7i / dy * jTg Vg d7^ /dy + . =0 J '^* aubstitating'the icalues of t from (32), \ji a»J:x + [;t a b]y + [b a c] z + + [r a ij = 0\ Cr a I x+ t b'Jj + ti b c] z + + I b 1^ , I (,gg) + (jcl]=o) ^adjx+^bcjy + l-c'Jz + Ihese are called normal equations, or better final ennations. They can be more briefly iiritten by sabstituting in ("a) the yalvies of the differential coefficients from (22). \wy^ =0, \vv£\ =0, Spva^ =0,... (24) Jf the weights are eqaal or unity ,7iU disappear as a factor, giving [a-] x+_gb]., + [ac] 2 + .... +[al] =0 l^ab] X +Lb'^] 7 + tb^z + ■ • • • + [bl^ =0 The solntion of (23), (24), or (25), will give definite valaes for x,y,2,.. which appligd to the approximate valaes t^.Y^i^, — will give the most probable ones irhich can be found from the given equations or obseryations. liinear equations can be arranged in the form of (22) without approxi - mate values ifhenevsr it will lessen the numerical ■.lork.the loss of hight er powers occurring in the reduction to linear form, and not in the later iiork. . 10. COSTaOti, NOSMAL EajAIIOSS. Jf in ( 22) we place a- + b^ + c- + ... Ij -.SI ^2" ''2 *°2 * •••• ^^ '2 and treat s similarly to 1, i.e., multiply each by its w a, and add the products; each by its tt- b and add; etc.; the terms of the first members ffill be the coefficients of the normal equations and the second members ■obeck terms for them, as below: [w a\] i [5 a 6] + ^ a c) + . . . . + \;r a y = ^ a sM {sab] +\7r b'-] +(5 b c] + .... + ^)r b 1] =(2 b s] V ( 26) [wac] +^ b c) +(» c^Q + :... + ^ a Q •[»■ c s^ ) Ex. I.Jordan,. Vermessungskunde, I, p. 35, gives barometer readings, as the means ^^TSyears meteorological observations, at 9 stations, as follcits: 1. Bruohsal,- h 12oT2 B = 75lTl8 6 . Heiden Ji^ 49^.4 B=7]fl. 13 2. Cannstatt 2^5.1 742.3,7 7. Ts-m 708.1 700.43 3. Stuttgart 270.6 733.50 3. Feeuden 733.5 697,64 4- ealff 347.6 731.27 9 Schop. 763.9 S9S.23 5. Preidrich 406.7 726.99 Plotting these valaes uith height h above sea level and barometer read- ing E as coordinates, the curve irlll be nearly or quite a straight line On this account Jordan assumes, B - X + hX, or X + h? T B = 7 (tbe theoretic function is a logarithmic one).- iasume X^ = 7S0'"'" . ?. "= - -03, and to equalize coefficients, ptt» h/ 100 (lOOy) = hV. Then LSA3T SCJA5BS. (%11..m.1. Table for Forir.ing the .loriial Sgaations. ■hlr a. ■ T 1 s V- X\ Vs 1 l.K) -0.83 + 1.40 1.44 -.93 1.63 a 2.25 -0.33 + 2.87 5.03 -.85 6.46 3 2.71 ^.15 + 3.53 7.34 -.11 9.65 4 3.48 ■10.92 + 5.40 18.11 +S.SO IS. 79 5 .4.07 ■fO.47 + S.54 16.56 +1.91 22.55 a 4.92 +3.45 + 8.37 24.21 12.05 41.38 7 7.08 +2.87 +10.95 50. 13' 30.32 77.53 a 7.34 +3.68 +12.03 53.88 27.01 83.23 9 7.59 +3.23 +11.95 59. 14' 25.07 91.90 9 40.74 +12.32 +52.03 229.87 37.34 357.97 9 z + 40.74 y' + 12.32 Check 62.06 ■40.74X+2S.37 y' +B7.34 =0 3S7.97 So\Mi-«^_x 1.7S; y' = -.695; y = -.00395; X X, + x 761.73; 1 Y„ + y -.08595. Substitating.the required equation 'oeooues, 3""= 76'i.7B"- .03395 tT. 11. M.S.E'S _ OF TBE OHKNOras. If in solving (25) the eliminatioa iras fully carried out, each untaoun .To.uld be finally expressed as a lin- ear function of 1, ,1,. .... and the E.s.e's. of the latter being the same as those of M, ,1/,,.... and kno/in, those of the former nould folios from S3. To effect this elimination use indeterminate Multipliers, i.e., mltiply the first of (25) by Q' the second by ff',..., and add the pro- ducts. -Then to find x, give such values to 9'.,C that in the sum or final, equation the coeffieients of the unkno.Tns shall be zero, ex- cept those of X rtioh shall be unity. This givss, C,o£ff>c- ■ -# l^a'-] 8' +jab] Gf +\aS\ S" +... . ** - — *, [able'. +5i^] ff' +[bc] ff" + *■ ~f [aclG,' +[bc] e' +b'] S" +'... - so that the sum equation reduces to ■(o.) X + [al] & + [bi] a' + [ell 9" + (bV The coefficients of the unkno/ins in (25) and (a) are the same. Hence if the yalues x.y,^; , are found from (25) in terns of 1,,1 ,...., those of a'.,af..., irould result from them by patting [al"! = ^ 1 h-fl [c5 = 0. This is also evident from ( b) . 'He no/f wish to sho-i that if c = m.s.e. or an observation of .leight unity t = a: ^ e of thn trai ,» ^f ., found from the normal equations.fhen, '^ a:..,.^. oi tno value of x, in (b) , x being a linear function of X +oil. .J + «lg . ^,g in rhioh by comparing ooefiLcients. ^ = o("= o<"= 3.^ er. agO-. + B, + c;"+ ... and (a>, dx =« 'A' -H»< "£»"-+■ D<"K"-v .. . . giving.since the sum of the products,6l (31) If the observations have different weights. they can be reduced to the same weight by multiplying by y/v, as in §11, giving t"'=Cf^J /(aim) ( 32) Having C or C. the m.s.e's.for the unknoirns can be found from §11. 13 SOIiOTIOH OP UOBMAL BQUATIONS.- The ordinary methods answer well when there are but few unknowns. Indeterminate multipliers are con- venient in special oases, while the method of successive approximation JO LEAST SWABES. (§14, Pig. 1, "ill often in70l7e the least labor. But the method of substitatioa.due to Gauss, ifill generally be found preferable ,as belo»: NORMAL EQOATIOSS. [aa] X +[ab] y +[ao3 2 + ... [ab] X + [bb] y + [bql z + ... [ao] X + (bo] y + [00] z + . . . Prom the first equatio«, [ab] [35] [all X = - — ;^- 2 - raal [aa) [aiQ Substituting, , , , prom the first of (a), ^ ^ ^j^_ ^-^^ ^j,^_ij g^ Sab^titating, fbO} "(JbHl L^b. 1] [oc.2]z + ... +[ol.2]=0 \^.Z\ (>■) where, ----- - - t[ari +[bl] + [cll = [asT = [bsl = [cs^] [asl fra'D = Q = Le«tfl = to.il-Jba^ Tbcii; rcl.2\ = rcl.^ -tbcflrti. ll fbb.il Bringing doira the first equation of each group.ne have the der ived nor- .jraj equations . CVaOt [aa) X +^_ab]y +[ac]z + •+[al] =0 [asl V [fcb.ily +^o.ll2 +..-iibl.l\ =0 [bs.ii 'f ^^^l fco.^z + ..jol.El = [os.2i J- 14. BOSK POS SOLOTIOH. 5 problem in astronomy is taken to also illusr irate the method of reducing a set of time transits, for clock error, az- imuth error and collimation error. The observed time of transit t, re quires: Correction for azimuth error, x, x si;i(^j^5 ) sec S = ■s«. (p-") ^^-j Correction for inclination telescope axis, i,- i cos (^-5' ) sec 07 +0.-a3 +0.52 +1.41 +1.00 +1.02 +2.51 -0.73 +0.75 +2.13 +1.01 +0.53 +0.63 +0.81 +1.02 +1.00 +1.02 +0.09 +1.27 - -7"5£» at ■to.ii ■•■0.51 ■H.i"l -0.13 ■fO.S3 +OSI +0.03 +sm ■^r.HI + 1.00 + 1.01 +%.1J I.OI + 1.01. 1.00 + 1.01. t^/i.\ +0.bi- o.ia -0..13 3,9 fc +I.«S -0,01 -0,13 -O.HH -0.19 +O.Tfe lot-l.'8fe + TL..9S +1..i"0 +a.Hl -s.ll ■m.i* +1,T1 +1,11 ■to.,SM +3. IV m.il °^a- .00 i- .Htl .no c.3oa .S33 1%I '.ooa S.b-Mt a.\. .099 - .^-SO • .S3I t.T0l -l.>'i-4- .i-MI .i%0 M.ll^ a.1. OHi -■ . 1 11 ■SS'iO -I.3T1. - .Oli" -.010 - .199 - .13i- + .oaa I109S .lie 1 ,100 + I.1S3 3,iiy + 1.0Si- + 1.1«3 + l,s« ♦ i,oin ISll rOi-^ w l,9«a 1,000 I, OHO 1,111 •H.S31 1,010 l,OMO l,QO0 1,010 ll.MQI .3n .l«0 • ,131 ► 10.^13 rl.OOM - .010 - .113 - .110 - .19b + .9(.i>' cliMST iS_ H.xlC l.KBO +1.1-S* +«.130 n.ltT n.ifc« +1.110 +l,i'Sl +3.9'il ,10.41« No^trvcCL E.q>A&, r a.i-1l-t--1,llb-M.+S.1TZ.- "..096 = J -H,1H>T(. +ll,H0T..^+'4.11l + '^"■'■ST =• L S.IT > +-^^ +io.oot- !.«' =0 -1,013 CKttV, + >10.1 So\-ut'lO'n. N O'TTn-cCV E.c^ca.\l.C)Tv5 Na r 3SL ■XL nr 30: SL UIIL 9, SHI ■+ lit -M,llt II. •HOT *,ll -l,*ll ll.MOT l9,S3i' i.yjfi- ■8.11 lO.Tti" g^ O^M _Q=3.. CWCVi. iS.m «,ii 10. -1*,09« .i4-.5jn - .013 1-3 .11 ^oV-u.tlOTL .401« l,i-4"fc g.XI -l.isso •A".3i"9 la-.ii'l lO.Ttfc -5.1«9 to. 6.i-PI .tfi'H .i>"\S J_ .101% Mil. • ,OOIH - OOb I/9.SHI nt-sM.i^b 9.S9<& 1,31 1 -I.St -.fcOi" HI.4M1 .008 13.11 4".1S-fc .i'ObT S.HSk -.fcOS' ,OlU -.113 -,fc06' .Oifll -,S"A"I .OOt - .<114 . ,6-»-| I 13.11% 1.131b- -11,91% la.lia .l%0 RcfrvgcyVs -I Is:•^c-i.■•^T) -I ■jnisc-io.Tis) hsSLL yT9 x = — oon Q.t.'ii.'Ts . -ST.voU-K ^= -,oon glMS.; ^=-.i-oi.-«; vM-vtV -..'.= -.'3i.-.fc, O.^^ .i-«;wvVK ■Z=-I.H<.Sfc, Q;^=. ,8190. . , . ,. , ^ X ■<»iiOn. -^=-««0, (!■,.= 1.36. u-e.O -113 & -wAr *t'\,t^vk.t!v ^T-O'm. *Vi.«. to-V -V .01 ■01 ■ "5. t =>/T.73x.029 sC". — M -V \J \jl .03\ .3*' .3* •'** ,001 .11 Oil's ,0i"% .O-J .001 .010 ^10 .OHO c ?V 1.36x.0t>SJ =0.20; 1 - vC^ff^rrOlS =0.13; 22r ^ .OlO .000 ,01i .OOP .031. — \; .04' .10 ,3y + M .03 .03 .1H ■33" 12 LEAST SOJARES. (SlS.Pig. 1, Clock correction, at 7 p.m., Oct. 2,1895.- -7" 52^0.82; Azimuth cor- rection = 41^.04 ± .23; collimatioD correction - -0?51 _± .13.- 15. INDEFSSDEST 0BSEa7.ATI0S3 BPOH BDEPENDEKT GOANTITIffiS. If there are m' equations, or as they are asaally- called, rigid conditions, connect- itfgthe m anknowns, the case can be redaced to §9 by eliminating nf an- knouns, leaving the remaining m m',. independeat until connected by ob - ser7atioQ equations. This metiiod is usually ased when d'. is small and for indirect obssryationsjwhen m' is large and the observations are di- rect the elimination by indeterminate multipliers will involve les^ la- bor as beloir. bet the m rigorous e^aations be, n\l,,\ V^^ = (34) iiiere V.J ,V,g ..... are the most provable values of the unknoiins. For each V. substitute the observed value if plus a correction v W ,-\i*f), expand by Taylor's theorem as in §g,and put ' df'/dMj ^ a J ; df/dKj hj... jdf /dUg- ag-.df/dKg ^.bg....; ffKj.Kg «-> ) Q ; giving ^I'l + ^£''2+ agVg t (q =0 bfi + bjVj+ bgVg + q, =0 (35) o,v, + c^v^ + «^»g.... +03 =0 These equations nusn; be rigorously satisfied by >r, , ,^ , ... The observation equations are'J 7, - li, = V. ; V^ \ =\; ■ •; or,Or, - v., NT, = v.VSjjlSTv- M»)W^ = »,Vvl,. The most probable corrections,! Vg, . . ., irill be those uhich make "1 ''l * "2 ^2 * '3 '3''' ^ minimum, or ' 1' 1 * 1* " e" 2*^^ 2 * ''^3''3^''3 * °° ^^^ This minimum is conditioned by (35). Differentiating, ajdvj + apdvg + ag dvg + =0 bjdVj^ + bgdvg + ^3'^ 3 * ^ =1*1 + =2<'''2 * V3 * '^ irjiich must be satisfied at the same time ffith (36) The number of these equations is i/.;the number of terns in (3S) is m ;' Snd as ffi> iii'.,!ie can find the values of m' differentials in terms of the B-fm'. others and substitute in (36). Ihe remaining differentials being c independent their coefficients sill separately equal zero. This elim- ination is effected by indeterminate multipliers; i.e., multiply the first e I/hj .ggj v^ =(agA + bgB + OgC ... J« 1/tCg from »hich 'v,j= k^ + y^, V.g = «g'' + Vg ,V.g = Mg + Vg . Since there sere a observations and m observed qaaotities.ifhile b'. gaantities have been eliminated, the difierenoe bet»een the nunber of oteervations and that of the anknosns is ie'.,so that (31)and (32) beooiie C^'[v'2)/!f . f'^&v^/ni' (40) 16. COHTROL. If in (37) we place a -+ b^ + c, + dj ^2 * '^2 ""=2 * ==2 and treat s the sSme as one of the other terms in deriving (38), the fol- loBJng checks will resalt. It should be noted that they do not contain the absolute terits as in(£Sl. ^a a/«J +^ b/w]' +[a c/ir] [a s/i] [a b/iT) +^b b/ir| +[b c/ii] - [b s/ir) (41) ^a c/tt) + [b c/ir] + \c c/»] [p s/i] . ,,^ . B qg- C ,3 : tc^^'^ "' (42) Similarly for independent observations upon independent qaantities, § 9, m'ultiply (22) by VTT .square and add, [irv2J=, \g sQx^+SfrabJxr + 2[rac3 xz + 2[»fal\ :j £wb2jy2 + 2[»:bo;] yz + i\v'bX^ y [rc'O z^ + 2[ircn z % (^23), ^ ^''^ CotVI = xlsail + yCirbll + z^" °S +[»1^'] ^^^' Applying (43) to tie example of gl4. [.v'-3= T 12,532 - 7.781 + 20.5«3 - .197 nearly- checlriag ty'"5 as found on tage 11. 17. EXAMPLE.. In the O.S.C. & Geodetic Surrey aeport,a8B0, 4.pp.8,are giv- «a ihe folloring differences of longitude.. Dates ObssE^ed Oi-t'feceoces Cor. 1851 Cambridge-Eangor O'*' 9" 23!baO 1(3.043 y-- 1357 Bangor-Calais a 00.315 ±0.015 vj 1865 ealais-ats.CoBteat 55 37.973 ±0.066 vg 1866 Hts.Coat.Foilh. 2 51 55.355 ±0.0*9 v^ •1S65 Foilhommer-Green. 41 33.336 ± 0.049 Vg 1672 Brest-ereeniiioh 17 57.593 iO.022 v^ '6 1872 Greennich-'Faris 9 21.000 ±0.033 v„ 1872 Ersst-Paris 27 18.512 ±0.027 v^ '8 1872 St. Pierre-Brest 3 26 44.810 ±0.027 Vg 1372 Camt.St. Pierre 59 48.608 •»0.021 v^q U LSA3T SOJARSS. 1359-70 Camb.-Diixhiry 1 1870 DMxbary-Brest 4 24 1867-72 Hasiiiagtoa-Cambrtdge 23 1872 Bashinfitoa-St. Pierre 1 23 Htmiber of conditions (34)or (35) I^-.+ 1= 14- 11 + 1= 4, (1= no. of obser- ved differences of longitade.n = no. pTstations) . (%lB,Pig.2. * ^ -'8 '.OSS ! '1—2 "3 '4" '5 -V, - v„+ v;+ v;+.049 = 7 -tv - 7 _ +.095 +7g-tVjQT.045=0 10 13. 14 ifeight s in7er3ely as the s-ouares o f th e uncertainties. UBTA \W7- 100/W = l/if a b c . d ^T bb/w bs/w 7^ .043 .18 -1 -1 .18 ..18 '2 .015 .02 -1 -1 .02 .02 '3 .065 .44 rl ^ .44 .44 "i .029 .08 -1 -1 .08 .08 "5 =.049 .24 -1 -1 .24 .24 "6 .022 .05 -1 +1 .05 "7 .027 .07 1 "a .033 .14 -4 -1 '9 .027 .07 1 -1 .07 "lO .021 .04 1 -1 1 .04 .04 "11 .022 -.05 1 "• "12 .047 .22 I 'IS .018 .03 1 , "IM .027 .0? ^1 -1 1.12 1.00 Normal ^ .25 A ;.05B -.086 i^l70- Bq.50.) FOSaeiON 09 BBOTIBED eaiSIITIBS. IS TheC for each observation can be foand by dividing C byVTF. XS. K.3.3. OF A FU8CTI0S OP THE RSOJIHED fflANTI^IES. For the case of indirect observations, the unknorfns being independent they can be ex- pressed in terms of the observed values as belo/i. "*, P = fCX.Y,...) = f(!!„4 i,Y„+7,...) = f(X,jr„...)^_dt' ,+ iSy'*- H + CjX + GjV +. .. ^^' '^^'. From §11, {c) , J -^q, y =-tpi],.. 3abstitating. ^3 c^ = tG.«'+ Gx(»"+. . . j'^i, +(Gp^+ Sg pr + ..)*ex^^- ^^^' The valaesof oCars gi/en in §11, (d); those of p soiild be found siailar- ly from the Q's. obtained by pitting 0,7lJJD, .. for the absolute terms of the normal equations, as in the probleiJ of ^J4; etc.. Bg.(44} can be transformed so that more of the numerical nork of solv- ing tne normal equations can be utilized, but the transformation is long and will be omitted. Por the case of direct observations let. e, - 4 v„ v^ .... V) ;There tne vs are cqnnected by (35),.i.e., ^1^1 *^fZ * ■■■ ■" Vm + b]k, Kbb] k,. ....t£>g: =0 ''" or using neights, [aa;;g k, f [ab/i^ k^+ Fag/n"] =0 [ab/n) k + [bb/vO k^+ ... [bg/»T =0 ^ 5°' These equations have the same coefficients as the normal equations (33) so that the values, of k can be easily found by adding a column of abso-' lute terms in the solution as in §14. Bx.l. Find the m.s.e. in a triangle side due to the ji.s.e's. of the ir.ea3 iired angles. Thefunction equation (45) is, P, « a = b silt i/sin B • b sin(«j^+ v^)/ sin (Mg+ Vg) 16 LEAST SeOARES (SSO, Fig. 2. g, df/d«j * a cot V^ : gx= df/dMg= a cot Kg (he; rigorous equation to be satisfied in closing the triangle is, A + B + C (l50+s) =0 .-. a,= a^= aj= 1, and (49) gives k^ "L^^Sl/CCD substituting in C47). s:'^ a'-'sin'- i((eVcV3)cot'- V., +ttV<'t<:.^t^ M^+iE^Jfe*! oot -a- cot -^ If ' t,=€i = Cj, , €*=^2/3a'- sin'- l"(oot'-Mi+ cot'' Mg+cot M,lto-tM^'»C'- If the triangle is equilateral, ti= 2/3 a%in2 r t'- [ the base has the m.s.e. t^.by S 3, t^ jould be increased by (a'-/b'')f^ ixiS, Find the m.s.e's. of the adjusted angles of a triangle in terics of If Sz: tbose of the measured ones. C H A P T E a .11. 1' H E R y. 19. PRIHCIFLS3 OF KOBABILITI. The mathematical probability of the oc- currence of an event is defined as the ratio of the number of nays it may happen to the total number of vfays in sliich it may either happen or fail;eaoh' being supposed independent and equally liable to Ojocur. Thtts if an lira contain a uhite balls, b black and o red ones; iurra single drai»l Probability of drajrlng a white ball = a/(a + b + c) [' ifif failing to drair a white ball = Ob * o)/(a + b + o> Giving sum of^probabilities = (a + b + c)/(a + b + c) 1 S(si1 Of drawing either a white ball or a black = (a + b)/(a + b + o) Of drawing a black.white, or red (a + b + c)/(a + b < Qf drawing a gresa bill = /(a + b + c) Se thus see tnat the probability is an abstract number which varies with the degree of confidence wtiioh can be placed in the occurrence of an e- vent, jzero denoting impossibility and unity certainty; that the probabil- ity of occurrence plus that of failure must always eoual unity;aad that (he f probability of the occurrence of an event which can happen in sever- cr[~lnde pendent ways is the sum of the separate probabilities. If a second urn contain a' white balls, b' black and o' red ones, the number of possible combinations or oases in a single draw from' each urn = fa + b + c)(a' + b' + o'),wiile the number of favorable cases for two white balls - aa'. Hence in two successive draws, one from each urn, Prob. of drawing 2 white balls = aa'/((a+b+o)(a',+b' +o'.)) (58) or by (51), equals the product of the separate probabilities. The same could be proved for any number of events. We thus see that the probability of a compound event, produced by the occurrence of severa} simple and independent events, equals the product of the separate probabilities. 20. PaOBABILITr CURy.B. With the accidental errors of observation the following axioms derived from ezperience .were- stated in %1: ' 1. Small errors occur more frequently, or are more probable than large ones. 2. Positive and negative errors of the same magnitude are equally prob- ablej«nd in a large number of observations are e^3ually frequent. '• V«ry large errors do not occurs Eq.53. ) P08M OP F(A) . 17 Prom the first axLom.it may- be assamed that the probability p,of an er- ror.^ ,is some fonctioa ot the error Prom the first a:xiOB,it may be assumed that the probability p,ot an er- ror ^,is some fonotion of the error, or P f( ^) (a) Praoiically there is a limit to the graduation and use o-f instrnments by irhioh t^ can hare only definite numerical values differing % the fin- est reading?' da ,so that the probability of an error-^is the ■{irobabil- ity that thi error lies between A and i»+ d& ,a tralue which iflll 7ary sith da,,', (a) woald be more correctly written p f(A )dii (53) Mathematically ue haye to treataas a continoous variable. Taking p as a continuous function of A, (53) represents a curve of the general form Pig. 8; for, by the first axiojt a- bove, small values of A must have the largest probabilities,? :by the sec- ond, the carve must be ^ytnmetrical about the axis of P: and by the third, p must be zero for all values of A greater than a- given limit i l.an impossibility ex- cept for 1 =°°. although it can be close- ly approximated. pi- .5 21. FORM OP f( A ).- Observations ^ nay be direct or indirect.i.e. , the observed quantities may be the required ones or they may be functions of them. Ss the first is but a special case of the second, only the latter need be considered. Let us take the observation equations of § 9. f(X.r ) M,= v, )■ (54) r(x,f.... ) - Kg there being n egiations and^unltno;7-n3 with n> m. The probability of the occurrence of a given series of errors, & ,A»,.. in II, .V^,... iri.ll be by (52) and (53) p f( A, )ai:i, f( A^) d/i^... (55) But the true values of X,!,... are unknown, and since A,,Av. are fournJ from them by substituting in (54), their true values are also unknoirn. The most probable values. ;rhich if the number of observations is great, may be taken as the true ones, of the errors and hence also of the unknowns, will be those irhich make p a maximum: or since log p varies with p.and the un- knowms are independent, eixcept as connected by the observations themselves, the derivatives of log p with reference to X,?... ..must equal zero. TVliq««,6WC« ^^g p . log f fA.) + .log f( A») + log dA.'flog d^,,*-- f . (A, ) dVay * f .( A J d'A/dY =0 [ ^' in which ---- - - -.^ f'X'A ) = df(A)/(f(A)dA) (57) 'Ihe number of these equations being the same as that of the unknowns, they will serve to determine them irhen f .( 4 ) is known. f(A) and f'.(A) being general. they must hold whatever the number ot unknowna When the number is one, the unknown is directly observed, giving for the errors, A,= X - Mj .ik^=!f -. Hg .... from which di/ dX - dft /dX .... - 1 and (58) reduces to ^ f .( A, ) * t'e Hi o*- cc»«-i be -founi f«v u '\n^\e \y {^foTn <\ %*\- t>f div-e.c'V obsch vB'JioT'S.a l\ eoiudW-^ "i'oi. Mokloij'T^*"' '*''^l>"nDplr»ori art i a^So )jTe tv*ucvo>ue f/)/ % IS LSASJ- seOARiS. (S23,Fig.4. or . traasposiag or X = (M^ + Mg + Mg + ...)/n (X - M^) +(X-Mg) + (X-Mg )... =0 i.e., A,+ A^+A,*--- = CompariQg tliis-;ritli (58), and remembering that each must hold nhatevsr the value of n, f .(A )/A a constant k ;. by (57) df(A ) / f(A) = ki d£i lategratiag •log t(AJ = iAVStlog C or f(/<,) _ c«>*'^'' io ffhioh e is the tase of the Naperian system of logapithms. Since as t(£») increases ,1^ diminishes, k must be essentially negative. As its value is unkno/in ?re may replace it by another unknovrn constant, i.e. .place k = - IM^ .giving p = f(^) d£.= e-^^'-'^dA (59"^ 23. C0N3TAST C. In deriving (53). the probability of an error between £. ;^ and A-)- d& .it iras assumed that p increased directly with HJi .which would be true tor small intervals. For larger intervals the probabil- ity varies ifith A, so that the sum of the separate probabilities would have to be taken. giving, pv - |\(^)dA C {^e-'^Vc^^'k^ (a) Since all errors are included between ±»>, th'e pro lability of an error between these limits ■ l.aad of an error betneeo aadoD{plu3 and minus errors being equally probable) = 1/2. :. 1/a C j1-''''"'^'dA If ^/(ZV) = t*'7d^ =''c\/2dt.4- t =» for Zi =0D,3nd Since the definite integral is independent of the variabl-e.we may also pat. 1/2 - CCv/5 C^e"*'' du giving '0 1/8 • 0'=t2|*|'"°e-t'-''*dtdu ( b) 'Ho integrate, take a surface of revolution gener- = t**i .or z - e"'!' Its differential volume above the plane PU. as found by dividing into elementary prisms, /lill be. 01. - zdtdu e"*'~^'' dtdu giving V' « sr c ^"^**^^ '**'^" ''^^ ItS' differential volume .as found by dividing the plane TiJ into ele - mentary rings of area = Z'rrxii, and erecting hollo« cylinders ol heights z.will be dtf. - Strrdrz a-rrrdre"""" , q-iui-nc^ V = -n-C'*^-»\j_.,i.T o> v) = -TrCe:^^Y^ =-"- (d) By (c) it is seen that the required integral = V./ 4, which by (d)=Tr/4. Substituting this value in (b). 1/8 C*C-n- /4. or C =ll/^)^jTr ,-.(59) becomss p. f (A)dA=aAe. /"ci/S^iT (60) 2S. VALUE Of ■pROBABIlITy ISTEGRAL bX SEBIES.-jSabstituting the value of v,^.M. E.q.62.; OEGRSS OH PRBCISIOIJ. 19 C in S22Ia). with the limits changed to ra and + a. or, with t!-/cxi^ t\ dA =£\^ d t.and the limits changed to -t = -A/c«lff) o-iA + 1 =A/corB;v ^ ■t^=i.y\=l>/^^) [J^^^ =(VVW^ lo^^^ Expanding e* by Maclaarin'''s theorem, e^= i+-»./i! +ie/x! -tV/a)* — '" €*= i-ty4'.+e-*--Cl/2) f ( «-»' /t'')dt = -(l/2t)e-'-'+ (-l/ZV)e-''%(l.3)/3'?|'(e7t"')dt fh" dt Ke-^Vst) (l 1^ l/C3tM+(1.3/(2e)* - l.S.S/CSf- )3*-.-) Bat fVV dt (%r^at-J^-*-\*-v=iFA-J^e:*-'aL't SiLbstltatiag, , ^ * {p)t 1 -Ce''M(l T V(2f^1 + 1.3/(21'-)'' -.l.S.5/(-2 t»)+)(62) Prom (61) and (62) Table VJI has been constructed from nhioh (t>\ can be found for any- value of t or ^/i.TSi In a given set of observations errors of different magnitude should oc^ car in proportion to their probabilities as found froin Table VII. This- gives a method of testing theory by practice, as below: in' the 18 inde - fsndently observed values for the angle Medniokea-Richsberg at station renk.givSn in Gradmessang in Ostpreussen.p.VS. Angle -V *v V* * A^q-V* -V ■*-\l v>- 33° St/ 3K25 -1.38 1.90 forward 49". 33 -8.50 40.74 7.50 -2.63 ' 5.92 ■«'.84 6.00 -1. 13 1. 23 «!• 30' 3.15 +1.71 2.92 4.77 ■fO.lD 0.01 4.57 «.30 0.09 3.75 ■H.12 LIS 4.75 ■10.12 0.01 0.25 •f4. 62^1. 31 6.50 -1.63 2.66 3.70 •H.:17 1.37 5.00 -0.13 0.02 8.14 -1.27 1.61 4.75 to. 12 0.01 4.04 ■K5.83 0.59 4.25 -to. 62 0.38 6.96 -2.09 4.37 5.25 -0.38 0.14 ^''^ 49,36 -8.-50 ■f7.84'"''"''* 37.59 rlO.69 IO,-7i 46.97 Kean = 83° 30' 34^87: £v"-] = 46.97 t is found 1,63 For probability of error 4 : t= p O.Ol; np • .2 1 With a larger number O'f observations a closer agreement would be ex- pected. ■' 24. DEGSES OF EBECISIOil. It should be noted that the value of p in 523 for a given value of a depends not on* tut on t =^/(tVT); so that in tiro sets of observations the probability of an error less than S in the first ,7ill be equal that of an error less thanS'in the second , j.f .a/tsff/t': e.g. ifC= ZC ,tlie probabilifcujif a^ err_or^ less than f in the 20 LEAST SQUASSS. (§i;&. jlg.4, first »ill be the aame 83 tiiat of one less taaaS'/? in tae second, dc tie probability of aa error less, thaa.say 1" in the first iri.ll be as great a3 that of one less than 2" in the second, or the degree of precision of the second is said to be only one-half as great as tnat of the first. The' degree of precision is then inversely as t,aad obser/ations can be rediiced to the same degree of precision, and their errors directly compared by dividing them by their corresponding cs •■" These quotients must in fact be abstract niinbers, since ^V(xt'-Us the exponent of e in (60). S3, COaSTMIt. In a large number of obser/ations errors of differ- ent yalues .vlll appear in proportion to their probabilities (as found to be nearly the case for a small number of observations in §23,. so that in n observations, or errors, there should be by (60) ndA' e /itVXrr) errors of value A', nd ik" e'^ciVirr) ° ' ' ^"< etc. Squaring each error, adding, and dividing by the total number n,we have for the average square the sum of a series of terms of the form. An dA e /(tVxfr ; and since the limits of A are mo, we Jiiii bave.ffith A'/xtv t'-and dA- t\f1'dt, ^ Average square L^'J/n 2tVi'7f je^Vdt 4£Vw[ e*Vdt Integrating by parts, .^ ,»» -v Substituting, ti^lM = t'' (63) fhich by comparison r7ith (1) shoss that the constant c of §£1 is the m.s.e. of §2. 26, AV.B8AGE ERROR. Similarly to §25, se have the ir.ean value of the errors taken Tiithout regard to sign. n =&=4/n 2tir2/lWj' e-»M3tT2VT£Tf^-i/? e-*'-?^ or A Cir2-it'''= .7979t. \ or £= 1.2533 n 1.2oSSB-ii]/" ) (^'^' 27. PROEAECiS ERROR, r. If a series of errors be arran^eil in order of magnitude, the central one is called the jsrobable error.' There thus be- ing as many errors with less values as .rith greater, the probability that any error taken at random sill be less than r sill be the same as that_ it is greater, and each equ.als one-half. Its value is found by placing p = 1/2 in §23. and solving for t(=Vtaa))^ ^""* A/tC|^,or r /(£ir?) = 0.47e3 from which r = 0.6745 t (65) The p.e. and lu.s.e. are botn ased in expressing the precision of obser- vations, ffl. GRAPHIC REFSS3ENIATI0B. If in (60), t- l/V^itxicix ..reduces A to t, p/(dt) f (t) = Tf\ e-'-'' from Tihich the curve f(t) of Fig.5 can be plotted by assuming values .fo^ t and solving for f(t),a3 belo.». Its general form flas sho.in in Fig. 3.. t f(t) t f(t) t f (t) 0.0 0.534 0.3 0.394 1.5 0.079 j .2 .542 .3 .297 2.0 .010 ^ .4 .43.1 1.0 .203 3.0 .000 Since p/dt is an ordinate, p, the probability of an error t. will be an arer * f(t)dt; while (p)t of S23, the probability of an error between' and t, will be the area from to t belpw the f(t) curve. Laying tnese values of (p)t off as ordinates for given values of t by Table VII. ss have toe curve £q.66.) RELATION BETWBJSB SBRORS. 131 (p)5. If A = t, t' 1/V2- 0.707 corresponding to the m.s.e. If A= .6745£, t" = .6745//^ 0.477. corresponding totBe T». e. . These ordinates are laid off at ab and cd; the latter will bisect theorea between the f(t) curve and the axes, and out the (p)* curve at the height 0.5, from the definition of p.e. The former will give the point of inflection of the f(t) oarve, for, plao- tOT second differential coefficient equal zero, d^f(t)/(dt'>) 4 t'-TT-i'i e'*' - 2if''e-*- '°'" t l/-\rS'= t', as auove. S. PRINCIFLB OF LEAST SHJAHES. In §21 ffe sa« that uith n unknowns dependent upon observation, their most probable values »ere those which ma Be p f( A, )dA, f( \)iAr. t{ A3 ; dAj-- a maxiiDum;or substituting the values of f( A^ ) t( A^ ) from (€0), p - c d A, d A^ d A^ ....)(.c: c;'er. -Hi^r^ ^c-^vxt-a a maxinium which since d'^.-dA^,....^ tv ■•• are constaots.orare knoiro from .tha observations, will be a maxiSium when (1/2) CAVE'-] is a minimuDi;i.e.,each error being divided by its m.s.e., or reduced to a standard degree of precisxon,§ 24, the most probable values of the an - Imowns will be those wbich make the sum of the squares of the quotients a minimum. Hence the name Least' Sou apes . If the degrees of precision are equal. £ can be factored out, leaving L^^3 a minimum. 'IfhenCA'^is a minimumnN^will also be a minimum, ^r § 5. ^i' ' n t\or CA'-J = Ev\l + aS^. ftit!/n S^= £^ = E^-'Q /n. Substituting, ^ _, /, ,, \_...-, Hence we may also say that each residual being divided by its m.s.e., ir reduced to a standard degree of precision'.the most probable values . of the unknowns will be those which make the sum of the squares a miDinun. We may also ndte that, since it was assumed as an aziojn that the arith- metic mean of a number of equally good observations is the most probable value, the .arithmetic mean must make the sum of the squares of the re- siduals a minimm. To test this, take some other value of the unknoFra as Xg *S . The residuals will be I'j = v- +S",Vg = v^S.t Squaring and '''"*• Ly«]=Cv^J*2SIy] *nS^ rtioh since \yi = 0. and nS^'is positive. will always be greater than [v^ 30. BELATIOH EETifEBN iVEBAGE.JlEAH SCUARE.AND PBOEABLE EBHOBS. To find the average error of S 26 in terms of the residuals v .with one aoknown. directly observed.we have Irom (66), v2 =((11 - D/nlLA'O ,'. it Diay'be concluded that on the average.^' aVit'' - B^n - 1), and A/v = v/n/(n - 1) or if 7 and A are added without regard to sign. 22 LEAST SaOARES (§31, Fig. 5, irro» (V) arrd (64) ^^^ ^^^^^ _j^--j^ ^ (,,) Su-bstituliing these values of p andTi^in (54), C =1.2638 t± 7)/V'n(n-.l) £,= 1.2533 [± \P] / n\/^^ (eS) vjhioli are kno^va S3 Petsr''s formlas. Prom (9) and (10). t = /t7»J/(n-l) ; y [7'5/«ff(^^ 1 BtoiD (65), __ r I f^-} r =.6745\V2/tt-});-/ f, .6745|r[7i]MB(nT:l) J ffbich are knoffji as Eessel's formulas. For the general case of inciireox observations, we 'aavs (31), t'=[v'g/(n-iii) shile, 1'-= M/n .•. as above (69) A /v. = v'n/(n-ii) ; [? A)/ [± v3 =^a/(Q-iii) T| =&Al/n [± 7y>/n(n-ni) \ (70) t- 1.25S3(±v1/«/n(n-iiiJ ; r = .8454^ 4^fl /x/n(n-iii) J mtiich are knowa as Iiaroth's formulas. For tbe ease of direct observations upon independent Quantities m' take; the place of n-m as in (4iJ. giving, (32) and i^of^h^!^'^ ^"^ ' ' "' •8464L*vV^ > ^^^^ t =V[T'y(n-in) and C =J"[v'yiii', with r =.6745C3 The values derived from the first powers of the residuals are often ase^ beoa-ose they are more easily computed: thev are ^not, hoifever. as accurate as those derived from the second powers. Heights are readily introduced, if desired. 31. DIKIT OF ACCDEACY. In deriving the preceeding formulas it has been been assamea ;(a) Ihat the number of oteervations is great; ( b) Ihat A can be regarded as a continuous variable; (o) That all constant errors have been elimnated. Bith but fe:v observations. (^ and (b) are only partially satisfied; still if (c) is satisfied, the oomputed m.s.e. will, on the average, be the true one. although in an individual case it may be somewhat in error. Hit as constant error is often present, the competed m.s.e. may be very mislead.ing.ualess the oiroumutances under which the observations were taken, or the reputation of the observer, are known. Again, when the number of observations.n, is great, an increase in n does not reduce the B.s.e. as rapidly as theory would indicate (£,=E/V"B), and finally there is in every species of observations an ultimate limit of accur- acy beyond which no mass of aocumlated observations can ever tenetrate. As stated by (fright (Adj. of Observations) " Experience, however, shows that in a long series of measurements we are never certain that our re- sult is nearer the truth than the smallest quantity the instrument will measure." In a word .re cannot measure what we cannot see". He then quotes from Pfr. Sogers. who found with the meridian circle the p.e. of a single com-" plete determination of the declination of a star = ± 0".36 and of the right ascension of an equatorial star i 0!026,who says: "If therefore the p.e. can be taken as a measure of the aoouracv of the ot=;ervatinns There ought to be po difficulty in obtaining from a moderate numbe? of It^ir- vat^ions the right ascension within 0!CS and the declination within OXZ. yet,^is doubtful, after continuous observations in all parts of the world for more than 3 centur5?,if there is a single star in the heavens whose absolute coordinates are known within these limits.) Ihe reason is that tne observations are not arranted so that constant error is sliada- ated.but only the accidental errors. £;q.71.) REJEOTIOS Of DOUBTFUL 0BSBR7ATI0HS. 23 In explanation o-t the statement that lire oanaot measure irjial ire cannot see". it may be said that the aniom l.Sl (small errors ocoar more |re - quentljr or are more probable than large onesT, applies only doirn to the limit of appreciation or measorejjent.and that below this limit another law of dis-tribution of error applies in which the m.s.e. of the mean does not increase as vir. 32. BBJEOTIOISI OP DOtJHTniL OBSEBTATIOHS. This is one of the most diffi- cult points in connection xith the ad jistment )of obserrations. An obser-t server is at liberty to arrange the observations and choose the condi- tions ander which he will observe as his experience and best jidgment nay dictate. Having began" the observations, if he finds the conditions anfavorable he is at liberty to stop, reject the uork already done. and bjgin again nnder more favorable auspices, fhen it comes to Individ - ual results in a set, if there is reason to suspect that an observation is' poor before obtaining the result. a note should be made to that ef - feet and a line drawn throagh the reSuJ-t, If the only reason for sus- peotiffg it is because it differs from the others. the young observer should hesitate about rejection unless the discrepancy is so .great. that a mistake is certain. The attitude of an observer should be that of perfect honesty and fairness, directing his effort each time to obtaining the best possible value of the quantity sought without be- ing biased by the preceeding results, and without rggard to them except to know in a general way that no great mistakes ^re being made. Having the different results together, and being familiar with the cir- cumstances ander which the observations were made, the observer can de- cide which if any he will 'leave out in making up the me^a. The computer in revising the work, usually assumes the right to revise the rejection of observations. For this purpose he, if not the obser- vfer,will usually require a criteridn. Several have been proposed. Peirce's is perhaps in most common use, but the follosing based upon Ta- ble VII has able advocates and is the simplest. If ^= 3£ in Table VOII, t = S/VT = g.lZ.giving p = .997; i.e. .only 3 errors in 1000 should ©"xceed 3 times the m.s.e. On this account, the criterion calls ifor rejecting errors greater than 3t in limited series of observations. Many object to any criterion, and leave the matter to the judgment of the observer, or to the computer in cases where more da- ta is obtained by subsequent observations or by. an advance in theoret- ical knowledge. See on this subject Wright, p 131-8 il . caAPTER. II r. APPLICAIXOH TO TRIANGOLATIOS. 33. TRIAKGUDATION. This is xlie most Bommon mettiod of obtaiaiog th6 true relative positioos of distant; points irhen considerable aooupaoy is desired. High poants wjien possible are chosen for stations or yertices.^nd signals are erected to make them iaterrisible. The horimntal angles betueea the signals are measured, and asaally the vertical also. One or more base lines are measared,»hioh alloirs of oompating all the other sides. TJia triangles are usaally solv- ed as plane by taking one-third the spherical excess of the triangle from each angle. . ^ . The latitude and longitade of one or more stations are obsenrea and the azimuth of one or more sides. The latitudes. longitudes and azi - jnuths can then be computed throughout the chain by focmilas developed in Fart. II. In adjusting these horizontal angles of a triangulation. there are two classes of errors or discrepancies ihich arise; one from the adjustueBt of the obserred angles at a station. the other from their ad jistneat in the triangulation. Strictly both should be considered together. but much labor is saved by adjusting the angles at a statio* first, and nith tjhese corrected values adjusting the angles of the'triangulation uithout reference to the first adjustment; and as the discrepancies in the first adjustment are small compared »ith those in the second. this method is usually chosen. 34. STATIOH ADJJBTMSHT. The adjustment of the angles at a station can be avoided by measuring the angles independently. and »ithout checks, This can be done by measuring, say the angles between adjacent stations. as in Fig.6,and using them directly in the second adjustment, or by meas- uring the angle from a reference line around to the right to ea.ch sta - lion as in Pig. 7. In the latter case each measured angle Bould cor- respond to a bearing -o-r direction of the line to its right, although for convenience the differences are sometimes treated as angles. In the first case, if the angles should close the horizon, the adjustment /lould reduce to dividing the discrepancy equally among the angles if of e - qual weight, or inversely as the weights, if the weights .are unequal. If instead of closing the horizon, the sum of all is measured, the discrepancy would be divided equally among the angles including the sum, if of equal »eight,or inversely as the weights if un - equal. The angles may be observed as in Pig. 8, SFinging from the left hand signal to each of the n7l others, then from the second to the. a-2 others. etc.j for nTlsets. giving a total of ain-ji)/2 angles between n stations. Denoting the observea values by W, ,V.^ , ,and the required ones by X, X, Z, or rather by X +x 'ia ■•■?. ^0+ z,the adjustment is readily effected by S9. Another method of measuring the angles at a station is, with circle fixed, to read upon each station in order to the right, then reverse the telescope and read in the reverse order. Other sets are taken in other positions of the cir- cle. The instrument arranged for this ?rork is called a direction instrument. and the method, the method of directions. Denote the- required directions of the signals, they make ivith the reference line, by lf,Z,U,.... or by Y + y, Z +z,a -lu,.. shere ^^q-Z^. ... are approximate values;alsd the angle beviieen the zero of the circle for each position and the reference' line by X'. + x'., 3f; + t(' , Then if the read- ings of the circle on 1 are Wy ,W, . ....on 2, ¥.' Ifif .etcthe observation equations will be or the angles which m; ■^-b Eq..72J 1VE:IG9TING. 25 y. f-I -M^T^.for llie first position r - r, = if"^ X* + r- Mg= 7^' , for the second position, etc. Or denoting the values of the first members nhea X„.Z ,are substi- tatea by l,as in?9, (22) becomes X'. ^1^. -.\ /. + y + I'g = ^'g ^'^l\ =v'i. f' + y + Tg 7'^ from which corrections can be foand as in S9. „>,,„»„„ CIS WBiaiTING. In §34 each W is the mean of quite a number of ob=,er7a- tiois;the m.s.e. of each can be' found from the separate observations bj- (13); the s(jiares of the reciprocals »ill give the .eights for .he ob»er- "wi?h these'??; m.s.e's. of the compited angles can be found .ith the val aes of the angles as in the problem of 514. If the angles are observed independently the m.s.e's. can be found by— (10) as abovej Saving tne adjSited angles or directions, the next step is to make up the triangles. In order to determine the number connecting different groups of points »e may note the following.' The number of lines reouired to connect p points with a closed fig- ure is p.and this gives one check uron the observed angles, we^y ad- ditional line will give an additional oheck.so that with 1 lines and P points. No. of angle checks i - p + 1 ^''^' This (ri.ll usually give ttie number of triangles; in exceptional oases, triangles cannot be found and polygons will have to be used instead flhen there is an excess in the number of triangles tne-oest shaped ones should usu411y be taken. ^ The triangle srrors can then be compit- ed by comparing the sums of the three an|les in each with ISO + spher- ical excess. Squaring these, adding and dividing ty the number of tri- angles will give the average square, or O for a triangle. Dividing by 3 will give the average C- for an angle, or by 3 the average t* for a di- rection'- by S3. Comparing this riith the average £"■ found for the ad- justed angles or directions at the station. and it will usually be found greater. The reason is that the former include only the observing er - rors, while the latter include both the observing and triangle errors . or those due to eccentricity of signal and instrument, lateral refrac- tion, one sided illumination, etc. Subtracting the former frqm the lat- ter will give the O- due to triangle error which must be regarded as con- stant, .adding this to the t* due to observing error for each angle we have the total for each «ngle; the weights for the triangle adjustment. »ill be proportional tcthese reciprocals. In case more than 3 of the adjusted angles' are required to form a tri- angle, the sum of the squares of the triangle errors should be divided by the total number of angles used, for the average V- for an angle; while in forming the sum of thetH for the adjusted angles at a station, each should be repeated as many times as the angle is used in different tri- angles and the total number of t'»us,ed as a divisor in obtaining the aver-i age. Polygons can be included with the triangles in following out this method for angles or dire6tions,if, there are not triangles enough to satisfy (72). The effect of the triangle error is to make the weights more nearly e- gual; if it is to be neglected, nearly as good results will be obtained v>^ neglecting weights as by taking them from the tS of the adjusted angles and with le ss labor. , ■'An angle is made up of the difference of two directions, | the same as by the difference of two^bearings. Thus the angle 1-2-3 = -1/2 + 3/2. where 1/2 and 3/2 denote the directions of the stations 1 and 3. '^ (a). The sum of the. angles in eaon jriangie ;"y„, : "r' gi;;.; igss 360' in each polygon, 180' times number of sides. + spherical excess, less ou U). The length of a side ifhioh cSin be found by computing through dit- erent triangles must have the same length by each. „„„i„d nn- (" e^ges rise to angle equations, (b) to side equations, both coming on £6 LEAST SOOARES. (S37,Pig.lj, Care should be taken to have the angles about equally well measured. 36 PIGORS ADJUSTMENT. The geometrical conditions to be satisfied in the ^Ur^ThrsSS oF'the angles in each triangle = 180- * spherical e|oes| or in each polygon! 180' times number of sides. + spherical excess, less 360 . (b). The length of a side which can be found by o .ferent triangles must have the same length b^^eaoh. d^ Thus in the following pentagon: Angle equations. There are 6 triangles besWes the station condition that the angles about t must remain equal to 360!- (35) becomes e^,^--— 7V^ • (a^) + (b^) + (f^) + q^ (■e^)"Ma.)"r(f^) -Vq^PO where, q, q,are the sums of the observed, or star tion adjusted angles, in the trian.gles.less 180° + spher-: ioal excess. or the triangle errors; and (a,), (b,) .... are the corrections to the angles. or the vs. Side equatiJ>ns, The triangles which give a side equation. or a check upon the length of a. side. will usually have one vertex in common. called a pole. while the sides radiating from it will each be common to two triangles. In maJcing up the check equation; the two radiating sides of .each trian- gle are written as a fraction. beginning with any one and |aking the ad- jacent ones in order in either direction around to the first again, the denoninator of the last can each time be taken for the numerator of the next when the last denominator will be the same as the first numerator, giving unity for the continued product. Each fTaction can be replaced by the ratio of the sines of the opposite angles in the same triangle, giving the required check on the angles. Thus 5 triangles have a common vertex at f, giving or. af bf b£ of sin b^ sin of gf gf ef c, Sin g. ef af sin - 1 «4 "" "b Taking logs sin a sin "2 ^^^ °3 sin 64 "" "5 log sin b -log sin a +log sin o log- sin b +log sin ■q„ t 1 1 ia }i a log Bin Cj + log sin e,, - log sin g^ + log ain ay - log sin e^- = The df /dMi,of ^5, = dClog sin bj/db, = Mod.cos b,/sin b,= Itod.oot B, where Mod. = the modulus of the common system of logarithms. d(log sin b,)/db, = ratio of change in log sin to change in arc, " d, /sinl" .wJiecB-d., = .tabular difference of log sin for 1". .", (3B) becomes d^(b^) - d^Ca^) ^ dgfc^) - d^(b2) * d^Cgg.) - d^CCg) «^(e^) - dg(g^)«g(a5) .- io %^ ^ "7 = ° there ?f = the value of the log sin equation^when the observed angles are substituted. * For convenience the decimal point is moved either six or seven places to the right for d and q. !_, ■ xXi m -■- 37. ADJUSTMENT OF' QUfDPILJiTEEAL. , Seneca take .1882. Angles observed independently. Weights found as in ?35. Spherical excess inappreciable. "C Triangle 34 Triangle 35 i/^ Bi'' 70° 27' 53'ro 1/w = 1.5 £."" 48° 53' 29.3 1/w =0.4 I,\; 60 19 06.5 0.6 Ml'' 60 41 56.9 =0.5 or 49 13 05.4 0.6 O.'.'" 70 24 31 .(I =0.6 180 00 4.9 179 59 67.2 6«C.1*.1 Triangle 36 41° 21' 25:8 l/w 1.1 29 25 52.7 - 1.0 60 19, 06.5 = 0.6 48 53 29.3 0.4 179 59 54.3 giTing sia.<. 3 angle equations ?tith q' +4':9.q' - -2" 8, q"' = - 5t7, e.Ct-u.o..viotv Po\>.o.tl.t tLt.N.,/L^OjUtOw/l-il«ilL«.'^'''-l'*''-'*=' '■ q.qioi-Mll I3tt '" q.sn » not a. V" '" 1.4- 3T.H To:vx«. ^o- N "-r-rrkcOV E.q>£\ ^.l.o-n-'i t^i-al V ■^ Ok V c a s «^ 'Vw '^^ ^W ''Ai ''^, '^>w ^. 'X. ^Vu, ii/w (Mil 1.1- 0.1 OA. O.H 0.4' OA I.I I.G 1 1 l_ r J 1 T r _i«j ll.S -31.1 -C4- iq.l •14.1 i.s- 0.& 0.-1 IJ4Jfl «q.4\ 33.14 Sl.».31 ' I31«.1< 13.11 1i-.6">. M.14- t4-1.s\" IJ4Ht 1.1 0.C K)J'? *.1l iJi- O.I 1.1 3.1 ■i.i ■11.11 -■7jOI WH.1 J.10I .1 1 S08MAL EQUATIONS CVutCK -0.39 D + 4.9 C +1.4 D - 2.8 C -11.U D - 5.7 C +?H«V.M D +275.0 These equations are acre readily solved with a sfliallfir ooeffioieat fos D in the fourtti. Tbas let D, = ICD .giving, 2.7 A + 0.6 1.5 E + 0.4 0.6 A 0.4 B + 3.1 0.39A +1.4 b - 11.11 :i:r + 7.01 - 2401.8 2.7 A 1.5 B 0.6 A 0.4 B -0.04A 0.14B from which A ^39) becomes HY) *0.6 C 0.04 +0,4 C + 0.14 +3.1 C - 1.11 -1.110 +24.11 D, + 4 0, - 2.8 D, - 5.7 D, 27.5 = -2.20. B =+1.52. = 1.6 (-2.20 7.5 (-.108)) ■= 0.6 (-2.20 +1.68) = 0.6 (-2.20+18.1 (-108))' (Nl') (Ml') (til") I'.S .1 1. 4 +2. 8 = (0i')-2.5 +18.1 (Mi'')*0.1 «11.8 (Hi')-l.O "23.9 -45.2 ' + 1.2 -67.9 - 208.0 -275.. 9 (nr) (o'i) (MJ^) should .108 (lV') ioe.o - q" These corrections applied to the obssrved angles aiil Si^e ine aMjasted ones.. 23 DSA3T SOaARaSv (■§ 41,Fig.l3, 38. HOJiBEa 4HD FOSMATIOH OF TBS S_IDS BOJATIOHS.. nen in any system the first two points are determined by the length of the line jqining them, the determinatioQ of any additional point requires t»o sides or t»o .directions so that in any system of p points we have to. determine p - 2 points, which requires 2(p - 2) directions, or by adding the first 2p - 3. Henoe in a sys- tem of 1 'Sides and p points. No. of side equations = 1 - 2p + 3 (73) where each side requires to be observed over from one end only. Stations betneen vhicb side equations exist foTin systems about a central point or pole including it in a triangle or polygon. Frequently the pole falls outside iihich makes no diiiereace in the sc lution. In either case there is one characteristic property; i.e., at every station three lines meet, save one.irbere p - 1 meet, there being p stations. Complications arise from systems vithin systems. It is less vork to take the pole where the least angles have been observed. in cases «hich permit of choice. In a oompleted quadrilateral nbere the angles are measured independently, it is best to take the pole at the vertex of the three triangles giving the angle*' equations; in other cases where the adjacent angles are used giv- ing 4 to a triangle the pole is conveniently takes at the intersection o.f the t»o diagonals. ,„„, The number of angle equations mas found in ("Z), each side requiring to be,.sighted .over in. both directions. ^ , , j ,. .^ ,. - ^ In a chain of triangles where two bases have been measured, both being ten tfarded. perfect. the, aDsolute term of the side equation becomes the ratio of the bases instead of unity. 89. ADJUSTMSST OF SSGOETDARV !F0 FRIUASK JHOBR. __ ■ adjusted by itself, the entire discrepancy would be _. ,ry. This would be accomplished by placing the corcsction to the adjusted or perfect angle, or its v. equal zero, so that the term containing it ffould disappear from (35)- Ihus in the following figure, we have given the an- gles of the -^^Ttmary triangle 1-2-3, and thosfe of the seoondary triangles i-X-i. 2-3-4, 3-4-i,de - rived as differences of direction. Angle equations K4/2) -(2/4) + (1/4) -(4/1) + q' •1(4/3) -(3/4) + (2/4) -(4/2) + q" =0 (4/1) -(1/4) + (3/4) -(4/3) + q"' = Side equations (l-4)/(2- 4) X (2- 4)/(3- 4) « (3- 4)/ (1 4) 1 or f d^(4/e) + dg(4/l) + d^(4/3) + d^(4/2) + dg(4/l) * dg(4/3)+ q"'= From these the corrections can be derived as usual. It a secondary chain connects at each end with a primary side and in manv other cases, the checks due to the connection are often brought in as a side The primary work having been thrown into the secondo.- equation .azimuth equation latitude equation, and longitude equation-thul :ith''fhe''priS:?fsidl''' °^ "' ''^' ^"''^*'' ^^'P'^'^^ll^l to,2Sd ^iioiding 40.'M-S.E. of iuy SIDE. In §18,Sx.l, it was found that ^= a^ sin'-l"(oot*A, + oot^ B, + cot A, cot B, )02/3 + t^ayb*- Similarly fop the next side. ' C= a^sin*-!" (cot''A^ + cot'-B^+ cot A^oot B^)«:V3 + £'-a^/73. ) ADJa3f.v.4NT Sit/IHIillJ 6A3SS. 29 In fig. 13 tie angles A and B are ased in oomDiitiafi tne side 5-8 from the base 1-2. The azimath of 5-3 can be oomouted from that of 1-2 b7 nsine onn It the angles C. Counting azimutii clockwise as usual, the azimuth of 1-4 Hould be found from that of 1-2 bv subtracting C, . The azimnth of 4-1 would differ from that of 1-4 b? 180° less the convergence of the merid- ians, and can be computed from formulas in Part II. The azimuth of m-i cjkiv be found from that of 4-1 by adding C;eto. If q^ = oomouted azimuth of 5-S less the observed or direct value. Azimuth equation a. -(Cj + (Cj - (CJ .... to =0 1 2 d "z Angle equations , b, UJ + (B ) + (C ) = 0; 0, (A J t (B ) + (C ) - 0, ^OT ^ ^Vxc^^T.!. 11 2 2 2 G'orming the normal equations as usual. ■nA-B + C-D... +q =0 -A + SB ^ = +A + 3C =0 Finding the value of B,C,etc..., in terms of A and substituting in the first equation, nA - 4/3 - 4/3 - .... + q = nA - nA/3 + q = z f- 3 q,/2n, B = - q^/2a, C - + q^/2n, D = - a^./2n, ce* Corrections (A ) = rq /2a,stitatia.]g ia (68), 30 LEAST SGUASES. (544, Pig. 13, (Aj = (2(1. + d„)A/3 (B J = -(d 4 2(l„)A/3. (C ) ^^ - U. - cl„)A/3 1 Ai Bi 1 A, B, 1 A, B, The oorreotions to the C angles will "tend to foot up zero, the differ- ences for 1 for the ft and B angles averaging abou't equal in a triangula- tion. The distarbanoe in th? azimuth adjustment rill thus be small . By calling the C oorreotions 2iero(d = d Jthe angle equations become, b (A ,) + (B J =0, c (A„) + (B ) =0,---- 11 lit Ci Normal equations', ( :dp. Cd^l) A . (d^- dgjB . (d^. dg)C ..... q, = ( d. -'d^ ) A + 28 =0 (a*^ - dg^) A + 20 =0 Prom which B = - (d *- d „)A/2, *-(d. + d„ )A/2 A, B, Aj Bv A = - ^-VL^r^^^A^B^'P (A ) = (d, * d „)A/2 ; (b; = - (d„ + d ) A/2;... 1 At 0| A| D| Shese corrections ifhea applied will not disturb the azimuth adjustment- BO that the length and direction of any line itill be the same computed from either end of the ohainv 43. ADjySTMSHT POB LATITQDS AND WSGITUDE. The observed latitude and longitude would not check throughout the chain due to local deflection of the plumb line. In joining new work to oli adjusted work at two points, as in filling in secondary triangulation-.the junction side computed through the new work must, be parallel to the old (azimuth equation), must have the same length (base line equation), and must oo'inoide in position at one end , which is best effeoted by a latitude and longitude equation-. This last can be introduced in the figure adjustment, but the discrepancy in good work will be so small that the equation can he omitted in the first ad- justment, and the error in latitude and in longitude distributed as in' a land survey without serious loss of accuracy. Bach station can then be reduced to center by the method given' in Part II, making the figure consistent throughout. 44. TSIGONOMSTHIC LEVSLING. There are three methods of determining the difference in level trigonometrioally; from non-simultaneous readings at the two stations; from simultaneous readings; and from readings at one of the stations only. Approximate formulas for the 3 cases are h, = k tan 1/Z{S^-S, ) + ( m i m ) k»-/2Rj a 1 h^ k tan- l/Z(Sy,-S, ) hi=kootS,+ (l - .2m,)k*-/2B, rhere k horizontal distairee; S, , S^= observed zenith distances; m .,ai„- ooefficients of refraction-; B^= radius of curvature of the arc join r ing the two stations. The m.s.e. for each result can be found as in §3, remembering that k is -well known, and that 5 is nearly 90°, C^f k''sin''l" tp2 + k''£^/3 a J (75) £i= k*- s in* 1" jJ/2 + k" e; /2Ri ( 7b) £i= k'sin*-!" £«- + k''c^/Ri' (77) In adjusting a net, the algebraic sum of the h's. in going around a tri- angle should = 0, giving for the number of the equations, the same as for the number of angle equations, 1 - p + 1. There will usually be enough reoiprocel observations so that the v-elue of m can be computed for the 1-ines observed at each station, Essigning ■;7eights to each reciprocal set by Eessel's empirical formula, D, n^/(n,+ n^), where n,,ii^, are the numbers of obser- vations forS;,S.. The weights to ,be given tc the differences in height would be the re- ciprocals of ithe i'y? fouiid above. Wright, p. 392, assumes £5= 2", e^= 0.C2, 83 being fair averages. 31 Sq'85.)- ADJOSTMSNT OP A COMPASS' S(JR/ar, -Ji 45. ADJUSTMSST OP A COMPASS' SUR7EY. For each si(l» the length and bearing are directly measured, ||^ile the latitude and departure are compited. The latitude equation is, L = 1 cos B (a) dL/dl cos B dL/dB - 1 aln B (b) §3, Ct= E^-cos^-B + Ejl'- 3in«B (c) Por the departure, D = 1 sin B (d) dD/dl = sin B dD/dB ■= 1 cos^ B (e) £o= CiSin»-B + C^'-co* B (tf) If ire assume as wa?: practically done by Dr.Bowditoh that Cj= ci'- = Ix constant =10 which reduces (lo) and (U) to £i = £fe = 1 C i.e. .the squares' of the ir..s.e'9i. in latitulde and in departure are each proportional to the lengths of the sldesv Jn §29 it is' sho»ji that for equal weights- the most probable corrections- will be those which make the sum of the squares a itinimumjand for une- T.ll "-Bifilits the sum of the squares- of the quotients found by dividing each correction by its m.s.e. Hence denoting the corrections in lati- tude xor the different aides by v^ ,v^,v^„ . . - , »/• /I, ♦ T> /I, * T> /I- * » mininMin. Differentiating, ' "■ ' * ^W/\ * ^\^\fK *^K,K,n, ^ =0 (60) The sum of the corrections must eqSil the total error in latituite wjth its sign changed, -q, ,a constant, *^ -f 7^^ + vi.>- + " q, Differentiating, dv^. ■+ dv^^ '+ dvi., + = (61) Comparing (80) and (81), and remembering that eac>i must hold whatever the number of sides, or v's., V,.. /I, = v..^/l, = vl,/1j = ^62) d.e. .the corrections. in latitude are proportional to the lengths- of the ■ Bides'.acoording to the Dr..Boiiditch rule. The same can be found, for the corrections in departure, giving, v=./l, =Vo,/l, =VoJl, ] ^^' If the corrections are required for computihg area,tJiey can be applied (directly to the values- in the latitude and departure columns-; but if they are required for a-geometrically consistent map ov record the correspond- ing corrections mast be found for the distances- and bearings-. This can be done by dividing the corrected departure by corrected latitude for the tangent of the corrected bearing, then dividing departure by sine and latitude by cosine for the corrected distance giving weight to t>\e value having the larger numerator, and using the other as- a chepk. This -required' the use of as- many decimal places as- the original Computation. Prom the differeQitial aquations ( b) and (e) ,the total correction to the side, dl = dL/cos- B t dD/sin B \ ,„,. = dL 1/L + dD 1/D J '■^^' The total correction to the bearing, rt3 = dL/D -I- dD/L, or in minutes-, dB". =- dL/D sin 1'. + dD/1, sin 1' (65) Equations (64) and (65) are readily computed with a slide rule, or even by inspection from the coordinate sheet. In eguBlion (78) an uncertain ty in c haining which would amount to 1 ft. in 500 wpuld give, .. t, = O.0447/r = V&UU C; or (in minutes) Substituting for different distances. Distance. Uncertainty in chaining. 10 feet 0.*14 feet 6D 0.32 100 0. 46 500 1.00 1 000 1.41 2 000 2.00 C = 0.044'? = 0.0447/\n' sin 1' Uncertainty in bearing. 0° 48' 21 15 07 05 03 An examinavion of these results shows that the assumption is fairly rea- sonable, although it gives too great weigjitto the "bearings of long lines and per too small to those of very sliort ones. 32 LEAST SaiASSS. (§43,?ig.^l4, _- 46. ADJ!JSTK?,NT OP A TRANSIT SURVSY. In an °rdiiary iransil survey no bearings are observed, but the horizontal angles belueen the lines are measurfd. In ooiputing- coordinates a meridian is observed or assumed and the bearings- found from the angles* To express these bearings in terras of the measured angles- in the adjustment equations, as should be done for 'aoouraoy, involves- too much labor. To use them as- observed quantities Hill give different bearings- and different ooor'dinates-, depending upon the direction taken around the fijBre for each in case the angles do not W fjl QH q" ', In ordinary work the m.swe.of an angle need, not exceed l minute, if care is- taken in setting over the points- and in plumbing the. flag poles. using tacte- on the stakes' for all lines- of less than 300 ft. .swinging without delay from the back sight to- the front sight, and lining in a "range"- point to suing from for all Unes- of less that 50 ft. Stith these precau- tions-, the m.s-.e. need not increase with the shortness of the line,a8' irlth ihe co-mpase- with which it is- a waste of time to guard against errors of eccentricity in setting up or flagging. On very rough ground, or in goinrig throu-gh brush, where the flag pole is partly- hidden, It may be difficult to keep the m.s.e. below 2 minutes-; while, for careful work,the m.s.e.can be readily kept within 1/2 minute. For good work the length of sight should be limited to about 1300 ft. It is- believed that tffe time reqaired to swing back by the lower motion and- forward by the upper for a second measure of the angle is- well re- paid by the freedom from, mistakes- and increased accuracy secured. Ordinarily, it will be -more difficult to measure distances to 1:500 than, angles- to minutes-.while an accuracy of 1:1 000 is seldom reached except on level groud or for city work. The aoouraoy of angle »ork is thus considerably greater than that of chaining, 1 minute in angle giving 0.15 ft. in 500 as compared with 1 ft. in chaiiiing;or,0.5 minute, 0. 15 ft.in 1 000, as compared with the 1 ft.dae to the more accurate chaining. On this- account it will be admissible to adjust the angles to close the figure (i.e. , so that the sum of the interior angles shall equal twice as- many right angles, less- four, as- the figure has- sides-) by distributing the .error equally- among the angles to the nearest 1/2 or 1/4 minute if they all been equally well measured, or* concentrAte the corrections somewhat npon.poorer angles- if not 'equally well measured. The bearings or azimuths- are then computed and assumed to- be correct in the final adjustment. This leaves- only the two oonditionsc Sum of latitudes- equal aero. Sum of departures- equal zero. That is-, 1,003- B, + 1,.008- B^ -t-ljcos B^ + = ^ /„-> l.sin- B, +. l^sin B^ + IjSin B, -i- = J ^°°' where the total corrections are to be applied upon the basis- of inacouracy in chaining. Denote the observed distances- by M,,M^,M3, .. ,and the required correct- ions by V, ,v»,v,, — . The corrected distances will be 1. = M, -f V, . 1^ = M^ -f v^ , 1, = M^ + v, . .. Substituting in (66), (iM, -t V, )cos- B, ■^ OMx ■^ Vi)cos (M, ••• V,) sin B, + {«» ••■ Vi)sin V, cos B, + v^ cos- Bi * V, cos- V, sin B, * Vx sin B» + v, sin where q, M.cos- B,* M^cos- B^-f U,oos Bj-v q^ M.sin Bi + MiSin B^* Hji Btor convenience change (67) to V, L,/l, + ViLj/l^ + VjLj/l, V, D,/l, + VxD,/li + ViD,/lj where L, ,Li,..., D,,Dt,., -.denote latitodes and departures. ' If C* for chaining increase as- l,or the weights- inversely as 1,(38) be* °°°''' Di-VflA * [CD/flB + q. = [L D/QA -f D3VaB + qt= ^ ^ ^, Solving. A (q»CL D/Q - q,B)VQ)/(lDVQ fcvg - t D/l) ) \ B ('q, & D/l] - qi[LVlJ)/(iDVQ [LVQ - & D/l]^ ) J (69) become, ,, = L.A + D,B Bk + («, + Vj)cos Bj =0 Bx + ('Ms * V, )sin Bj =0 ,or B, + q, = \ + .q.= f B^ ;os Bj-v = error in latitude, in B3+ = error in departure. +q, = \ +q.= o I (67) (68) (89) Vi = LvA + 5vB_ > (90) Adding, 1V)'= A'"[l!) + B ioi =0, nearly Also, V. = V, L,/l, = A [i'/l, -f E L,D,/1, Sq.96.) ADJUSTMSHT OP A TRANSIT SOHTSY". 33 ^B, = '. D,/l, = A D,L,/1, + B D;/1, irilh Lrj = - q,^nd L»o"3 = - Qv If tke Inacodracr in chainiag increases' directly with the distance (C rarying as l),or the weigtits' inversely as 1*,(Q8) become, Ipik * [L p\ B + q, = -1 .L a A, } (91) (02) \ ^ + LDOB + qi =0 ^. wHh A = (.qv[L 0| - q,[fi^ )7V\P^ O?!! - Cl 61 ) B - (.q, [L g - qxLLT )/(CD'1 M - [I ^ ) 7, :^ L, 1. A + D, 1, B »4 = L^l^A + D^liB In order to equalize muibers' ao as to retain.th^- sane number of decimal places throaghciit.lOO 1 iS' used in place of 1 la (e9),]iiaUng the values' of A and B 100 times- too great and requiring the values of v to be divid*- ed by 100. If it is assumed that the error in chalsing increases directly vith the distance, (78)- may be changed to t^ • ej. - 1 J< constant 1 irhioh changes {1^) to Ct = «?b " 1*0 (60) to dv. V. /l^ + dv..j /I* + dv V. /l^ + = (62) to v,;/l; = vv^/]i''= v.^/15'-' ' (83) to Vo. /ir =Vo,/IV =v„,/i; 1.9. , the corrections in latitude are proportional to the squares of the sldas'.as' also for the corrections in departure. An ezamination of (83) shoirS' that an error Of 1:500 in distance Kill give 1/500 = £^1. or Cb= 1/5OOX.00029 =7', or for 1/1000, £'e= 3.5'.. These ratios are more reasonable for transit ifprk than those tabulated from (re), but it irould require an accuracy of l/lO 000 in chaining.or the best grade of level groono city irork to reduce the oorrespondeng angle error to a value easily attained in ordinary transit work^unless the fig- ure has- a very large number of sides. In this method the error of closure of the angles ifoald first have to be distributed before computing the coordinates. Example l.The following field measurements' »er)9 made ifith transit and tape: Sta.l,44'33.8'Ji, 887.24 ft.;sta. 2,8'-04'R, 451.75 ft.;sta.3,12ffl7.5'R 921.80 ft.;sta.4,89*28'P,212 ft.;sta.5,2°35.5'L,317.3 ft.;sta.6.91»l 443.6 ft. The deflections- foot up 380* requiring no adjustment for angle closare. The line 6-1 is- nearly north and south and it is- taken for the meridian. In computing the coordinates- ooIums- are added for LVlOO l.DVlOO 1, L D/lOO l.made up »ith slide rale from the distances- and coordiaates as- 9,5' eire n beloifT Sta- cion Bearing. Dis- tanoe Latitude, L. Departure, D. L* D'- ife^l -t- — -»■ , — 100 1 100 1 1 2 3 4 5 8 N 44° 38. 5' E !J 52 42.5 E 3 2 00 ir N 88 34 » 3 88 50.5 W ITorth Totals 287.24 451.75 921.60 211 317.3 443.6 304.37 273.70 5.30 MHJ.bO 921.04 6.41 201.83 359.40 32.18 211.03 317.24 1.45 1.65 9.20 0.00 0.01 4.44 1.42 2.85 0.01 2.11 3.17 0.00 1.44 2.13 0.32 -O.05 0.08 0.00 923.97 927.45 923.97 561.23 561.33 581. 23- 16.75 9.58 3-95 - 0.48 -0.10 A = ('-0.10x3.95 + 0.4S>c9.56)/('iaO - 15.55) = * 0.029 B = ('-0.48x3.95 * 0.10x16. 75) /(ISO - 15.55) = - 0.00.1 V, = *ja.05 Vl, = ■•■ 0.04 Vo. = + 0.04 V,. = -HD.OS V, = - 0. 23 Vh= 0.00 Vy= 0.00 Vt - + 0.13 W = o«oo + 0.05 vlJ - ■► 0.28 vt, " 0.00 vr^.= '.00 Vt,^ - + 0.13 »D, = * 0'.08 »», = * 0.01 »h, " o.x 'ol- 0.00 = 0.00 - qx LvlI = + 0.48 - q, lVo3 = -t 0. 11 If anyline iS' regarded as perfect, aS' in connecting »ith a survey alf ready adjusted, the corresponding correction is made zero and the corres-> ponding L"'/1Q0 1,D*/100 1, and L D/lOO 1 omitted in the summation for A and B. FftBT II. GSODBSr CftAPTHIB Iv INTRODOCTIOH 1 GEODETIC saWEY. Geodesy is^ the soiea-ce aad art of nakiBg tke ■easireBeats and redactions, regaired in- relatwely^ looating.^itli aocara- or oh the earth's, surface, points »hioh may be wj-dely separated. It hence supposes a knowledge of the fignre of the earth.of the rarioas. Pfenomeaa, «tioh effect physical measarements and of the oonstraction and ase of in- stranents.in addition to the accaraoy of sight and tonoh so charaoteristio of the good obserrer. ,, , , ^- ■ ^ A triangulation-net.or chain of trian-gles., is asually employed as giTing the best resnlts-, both in quantity and accuracy', for the expenditure. Ble- irated points, are chosen for the triangle vertices, at distances, apart 7ary- ing itith the character of the survey from a feir; miles. a.p to a hundred.one or more level lines, shorter than the others, are selected for base-lines, in' such positions that they can be readily connected tith the nain' net; signals, are established niiich define the vertices accurately, yet are conspicuous' enough to be seen by the aid of a telescope from the adjacent stations.; tbe horizontal angles, of the triangles, and usually the vertical also, or the inclinations of the sides.are then accurately measured Fith a theodolite, and the base-lines, irith a base apparatus. All the triangle sides' and the differences in elevation of the vertices- can then be computed. Usually the elevations above sea-level of one or more vertices are Mess^ ared;Hhile astronomical observations are taken to determine the latitudes., and the distances, in longitude from some observatory or reference station- of one or more vertioes.,and the azimuths of one or more sides.. The actual positions, on the earth's surface, both horizontally and verti- cally, can then be computed. The objects, of a geodetic survey are usually tspfold: (ta). The location or recovery of boundary and. division lihes or noBuientSi and the furnishing of a net »ith »tiioh to connect a topographic or hydro- graphic survey so that the inaccuracies of the latter cannot accumulate over large areas.. 0*). The accurate determination of the figure of the earth. The distance bet»een the parallels or meridians through any t».p stations or vertices, results, from the triangulation.and their difference in latitude andTlongi- tude.from astronomical observations. Dividing the difference in latitude in linear units, by the angle in degree measure, or in -n"=measttre nill give the length of a degree, or the radius at _carvature,of the jneridian. Prom these values, in different latitudes, the semi-axes, a and b, the meridian quadrant Q, or the semi-major axis, a and the eccentricity e or ellipticitr . : , %^^'" *"'*' °^° ^ computed, assuming the section an ellipse. or the actual form can be approximated. iimilarly , th* parallels can be oomiat- ed,assuming oircles.;rgiving an ellipsoid of revolttioa,-or their actsal' shape can be approximated. <»«-«wj. 2 HISTOHIO oaiLINB. (b). In glancing at the development of the science of geodesy wp may note as of special interest: "'-•."nm h, ^i-t"/*' aotjenticated hypothesis, of the spherical form of the earth ^ „Pythagoras.,irho is. supposed to have been born about 53g B.B HP nnffifif^L fJ"!^;!4°'^°^ ^"^ oiroumferehce by Eratosthenes, 230 B.C. oppfif^?^*® ?''® ^^1^°^ °^ deducing the size of the earth from a meas- ured meridional a?o for he .found that rtile the sun's rays »ere verti! oal at noon during the summer solstice at Syeue in southern Sftvctthev^ cir- ra^olQtxons- made by a carnage "J^f ^^". ,f"°^ .i,^ias by aa unasaal and redacing the broken line to ^»e ^eridian ^i^i-- ^ ^ eoBpengartoH -Of errors.a ooBpatea oircanferenoe only '*''® '^^^ /, v,„ qn^llias of Holland.1615.it being the first in •The arc measared by Snellias .01 "=j:,^°°"' , '. gg jged 33 triangles^ ,i,ich the principle Of ^^^f f ^^^^°? "f ^^'fe^^ih a sector ha.ing sights. .Leasntfed his base-line vith a f ^^^'^f ^??ffi<.^r. ais. compnted oir - 3t,taohed;and foand a iBendional arc of about 1 n .. casiferenoe n^s 3.4% too small. tplescope and its adaptation to The introdnction =* .°^°f '^^"Ig" ^ eSlfa triangalation over an ^U^^rr^^f^ n|^3 yen -lesoong^nd^ ^...ed If ^'Llir^ll lToll'T^'\ll^rjZ% ..Lh alelescope »as at- tached. The faotS' reported by Richer. on his- retarn- frooi an astronomical ex r pedxtion in- 1672. yiz.that his- clock, njiich beat seconds.- at Paris^ be - fore starting, lost aboat t»p minnteS' per daywjiile at the island of Cayenne, 3. Amerijca, and could only be corrected by shortening tha pen - dalam' 1 1/4 PariS' lines-. The annonncement by Hewton, Pi'incipia,1687.of the theory of iiniyersal gray itation, and of the corollary of the oblate spheroidal forin of the earth. The' first »5is confirmed by Pioard'S more accurate degree-length; forwith the diameter of the earth thus giyen.the force of gravity at the surface and the force required to hold the moon in its orbit, »ere to each other inversely as the squares- of the distances from the earth's center. TJie second w^s- confirmed by the behavior of .Bichef's pendulum; for sSioe by Chorch's-'Mechanics, §78, (irjiere t is- the time of oscillation in seconds in vacuo; 1 the length; g the acceleration of gravity; and h the versed sine of the semiarc of os- cillation, supposed small) an increase in' t for a given value of 1 in the lov^r latitude, indicate^-a 'decrease in g or an increase in distance from the earth's center in- approaching the equator. The extension of- Pioard'S triangulation each way from the vicinity of PariS' to include a meridional arc of 3° 31''.,bet»pen 1683 and 1716 by J. and' B. Cassini;from irjiich the length of a degree of the neridiaa das- found to be less at the northern end than at the southern. The earth wpuld thus be a prolate spheroid, and not an oblate, as advocated by Sew- ton,llaygenS',and others. Buy gens had published in 1691 the results- of ezperimeatS'Mihereby he found that a flexible hoop *hen rotated about one of its diameters wpuld become flattened at .the poles- if unrestrain- ed. The controversy which aros.e finally induced the ffrentsh Aoademy,- ^S' the French at thiS' t^me'took the lead in Geodesy, -to send ou.t tup ex- peditionS',one to Peru, uniler the equator in 1735, the other to Lapland- under the Arctic circle lo' 1736, to definitely settle the giiestion. The degree length in Lapland.when made knoirp in 1737,ii;as found to be gre'.oit- er than at Paris; Cassinl'S arc ifben revised in 1744, gave a greater len'gth for a degree of the meridian at the northern end than at the southern; so that /rhen the result from Peru was received abou.t a year later all agreed in confirming the oblate hypothesis. The details, of the measures- of these arcs- are extremely interesting. The first is' de- scribed by Baupertius in La Figure de la Terre, Paris, 1738, and fey Oir- tfeier in Journal d'un Voyage au Herd en 1735 -!, while its- remeasnre by Sranberg, 1801-3, is described in Exposition des- Operations faites- eir Lapponie,. par J.Svanberg,Stookholm,lS05." !£he second, by Cass-iii de Ihury, in La meridienne de I'Oteervatorie de Paris'.veriffeex, Paris., 1744. And the third in La figure de la terre, par M.Bon4aer,Paris.,1749,and Us- siare,des. trois pramisrs Degres da. IJeridien par M. de la Condamine.Par-* is, 1751. Clarke, Geodesy - 0iford.JS80.pp.3-13, gives an excellent resf .ujne of the wsrk in Itsplsna and Pers.. BlSIOaiC OTItilHE ^ ^ , ^ . 3 The trlaagdlatioB to ooanect the obsfr»atsr»es of Pana and Green - nioh proposed 1798; and' that to determine the earth'^ meridian qaadrant, 1791,frorthe measare of an arc of aboat 9' 43'. extending soath from the ertremt northern end of Pranoe.one ten-millionth part of this quad- rant »a3. to be ased as. a standard nnif of length- to be called a "^ter The French, introdaced the repeating circle (see ^S4) on the first and the Borda base apparatus^ (sea |5g) op the seoond. .ifith the one, the angle to be measured between two signals, is. added on the oireie as- many times, as desired, or as. there are repetitions, -as may be done nith an or- dinary railroad transit, -iihen, subtracting the initial reading from the final, iiith 360" added for each fall ciroamference passed, and diTiding by the number of repetitions, the Talue of the angle is. found -with the er- rors, of graduation and of reading divided by the number of repetitions., or br as great a nnmber as desired. With the other, the change in length of the measuring rod" due to a change in temperature is. inferred from the actual change nith reference to a companion rod haying a different rate of expansion'., forming a metallic.or Borda, thermometer, ilhile the theoret- ic adyantageS' haye nerer been fully realized in either case, the Importance of the principles, deyeloped may be inferred from the fact that both" haye held an important place in geodetic vprk from th^t time to the present. Sor descriptions, jo^, the French portibns- of the irork see Expose des- Oper- ations faites. en France en 1737 pour la jonotion des Obserratories de Paris- et Gren«ich,by Um. 8asstni,Uechain,and Legeqdre;and the three yoI> unes. entitled Base du. systeme m&trigae decimale.by Delambre, Paris, 130@t13. On the part of the triangulation irhich fell to the English, a. Bamsden theodolite v^s- introduced, of such excellent quality that the re-' peating circle, and the corresponding method of repeating angles, has- ney> er crossed the Ohannel. This- instrnment has remained in use, on primjiry triangulation in England and in -India to the present time;and Col. Clarke, in 1880 (Geodesy, p. 14) says, that irith the exception of some yery trifling re.pairs.,it is as good as- »hen first used. The circle, 36 inches, in diame- ter, /las graduated uith a diyiding engine by io« into spaces of 15'j it is- read by three micrometer microscopes to single seconds-. The telescope has a focal liength. of 3Q inches.,and is- supported by an axis tup feet long. Ebr a description' of the oork see,Accoan't of the Obseryations and Calcu^: lations of the Principal Triangulation-... by Capt. A. B. Clarke, R.E., London, 1358. 3. aiSTORIC OOTLINS. (lb). The increased aoouraoy introduced by the French and English on the suryey to connect Paris- and Greenirich.and on the. suryey to determine the length of the meter, mark the close of the eigh- teenth century as- the beginning pf. the -era of modern geodesy. General interest 'in the subject became anakened and geodetic suryeys be- gan to extend oyer Europe; iihil« the degree of accuracy- attainid, in some respects- at least, compares- not unfayorably with that of the present time, E.G., large triangles irere easily closed jjithin 3" uith the.3a-inoh Rams- den theodolite; a maximum limit irhich, >\a& long been prescribed by the d. S. Coast Siiryey for ppimapy triangles, althouga the ay.erags error .ta ySrr much less. In England, the Ordnance Suryey deyeloped from the triangulation ujn- necting Paris and ereen.;fioh;it has extended oyer the entire kingdom ■^ith a triangulation and detailed topography, under Gen. Roy, Capt. Mudge, Ool. Colby, and Gen. James, res-pectiyely aS directors. See account of the Trigonometrical Snryey of England and Kales, 1T99,, also Aocouat of the Obseryations and Calculations of .the Principal Triangulation., by Cant A.R.Clarke, London, 1353. <' v In India, ffork sas commenced in 180S under Col.Lambton,- a short arc Tt^hJf^hSE®'^ in 1790 by Burro*. (Montliohe Correspondenz XII, 433) - • It has been continued under Col. Bvgrest,3ir ■'.(augh.tiieut.Gen.flalker and p2^™ii^^^®''!L Tile, objects haye been mainly topographic, but in order to properly check the irork oyer such large' ireasTchaiis of primary Iri- an gles,,»xth an occasional tie-chain, at right angles have beep carried along meridian lines at such distances apart that the interyening country can readily be coyered by secondary triangles. A meridional arc of about £3° 49'. has- resulted, and an arc 'of the parallel of some :30°; the first is of yalae in degree determination^ but the difference in longitude has not .been determined with sufficient accuracy to warrant the use of the second. 4 GEODESy. 64 See.Aa AoSoont of the Measureaient of an Arc of the Keridiaa betngen the Parallels, of 18" 03' and 24° 07',, ,by Col. Everest, London, 1830;also An Ao- ooant of the Measurement of fno Sections of the Meridional Arc of India.i. by Dieat4Col. Everest, 1347; and Account of the Great Trigoaonetrio Sarver of India.by Lieut Gen.ralker to 7ol.3I,and ander the order of Ool.Thaillier from ifols.X to nV. in' 1390,inclasi7e. On the Continent, geodetic »ork .W33 begun in Prussia in 1802,by von Zach. In Siritzerland and Italy sork i.as begun in lSll,the object be- ing to join the French Triangulation and secure an aro of the ^ral- lel from the Atlantic Ocean to the Adriatic sea; ».hen completed in 1832 it if:as not found very satis factory 'and has never received much credit. In Russia, the first nork of value vras begun in 1317 under Tenner and Struve; in 1355 a meridional arc of about 25° 20'., extending from the Danube to the Sorth Sea,had been completed. Tbe report of the nork in the tuo volumes, Arc du Keridien,de 15° 30'. entre le Danube et la mer glaoiale mesure depuis 1316, jusqu'en lS55;0uvrage- composs sur les differ- ents materiauE et T'edige,par P.C.U. 3truve,3t.Petersburg,lS30,is consid- ered the greatest contribution yet made to the subject of the figure of the earth, and should be studied by all iito are interested ia geodesy. In flanov e.f , Gauss measured a -meridional arc for a degree measure, 1821 - 23, and extended the triangulation over the country, 1B24-44. His ifork iS' classic;to it is due the first application of the method of' least squares in the adjustment of a triangulation net; the theory of conical coordinates; the general theory of geodetic lines ' on carved surfaces ; and the invention and use of the heliotrope. In lS31,6essel and Bayer, began a triangulation to connect the chains of France, Hanover, Denmark, Prussia and Bavaria, »ith that of Russia, and to 5erve for degree-measurements. This »ork is also classic: the publi- cation of the report, Gradmessung in Ostpreussen and ihre 7erbindung,,,by F.W. Eessel, Berlin, 1333, is thought by Col. Clarke to mark an era. in the science of Geodesy, on account of the precision of the book,and of the Bork of ithich it treats;maay of the methods »hich are there for the first time described being' still in use. The Russian and Austrian chains nere connected between 1347 and 1351; and the Siiss and Lombardian chains at about the same time. The English and Belgian sere joined ia 1361. About 1333 the Pemaneote Commission dec International Erdmessung,-The International Geodetic Association, -was organized largely through the efforts of Gen.Baeyer,Bessel*s oolaborer. Ffr.Helmert of Berlin, is direc- tor and A.'dirsoh, of Nareoiburg, permanent secretary. For an account of the recent »ork in Europe, reference may be had to the yearly reports of this Association, xhich includes some tvientyrfour countries. But little »prk was done in Italy until the formation of the Italian Com- mission, 1363. Work /i:3s begun in Spain in lSS3,and excellent results have been obtained under Col.Ibanez. A remeasure of the French arc of Oelambre and Mechain vras begun in 1370 under the direction of U.Perrier, and this sas toUoued by an extension of the French and Spanish chain across the Ksd- itetranean to Algiers in 1379, giving a meridional aro of 27° extending from the Shetland Islands to the desert of Sahara. The chains of Russia and England have just been connected through Central Prussia nith small discrepancies between the ten base-lines joined. Accurate topographic surveys and lines of geodetic levels have also bean extended over the greater part of Europe. The develojiment of least squares has added much to the precision of geodetic work. The theory was first stated by Legendre in 1305; 'it was added to by Adrian in 1803; but its full developement was due to Gauss in 1309, and its first application to the adjustment of a triangu- lation was made by him in adjusting the Hanover arc as already noted. The method as now extended and perfeqted is applied in the reduction of every important geodetic survey. 4. GEODETIC WORK IN THE OHITgD STATES. The English Astronomers, Kas. on and Dixon, in running out the celebrated line bearing their name, found the position of the division line between karyland and Delaware which co- incides approximately .lith the meridian to be on low and level ground, and .hence well adapted co direct Esao-ireiieat tor a degree determination, 'ao- li . T.RIAN.5JUTI0N, S oordiagly.nitli the aid of the Boyal Society of Lonaba/they made a direct measaremeat nith wooden rods, starting at the soath-»est cornej? of Dela - »are and extending into Pennsylvania, of atxjiit 1° 29'.,and determined the az- imuths of the different portions of the line and the latitadesof it^ ex- tremities. The iTorS, described in London Philosophical TranSactioas, 1763, by tiasoa and t{asksliae,is not accepted with much bonfidenoe. The U.S. Coast Survey »as authorized by" Congress, in 1807; but, oiring to lack ot funds, Aork nas not commenced until JS17,and but little sas done ex- cept in detached surveys along the coast, until 1832. The triangal-atioa , nMich "as commenced in the vicinity of Sea iari Harbor, has been graclually extended along the entire Atlantic boast, along the Gulf coast and along the greater part of the Pacific coast, not including Alaska, In 1371, tiie project was authorized of connecting the Atlantic and Pacific systems and of furnishing trigonometiic surveys- to such states as should make the. necessary provision for carrying on the topographic and geologic por- ions of the »prk. , , ^ ^v ^J- ^ he transcontinental chain, jrhioh extends approximately along the thirty- ninth parallel, jas soon begun and is noT completed, C13991giving. an arc of about 28° in latitude, and of about 49° in longitude. The opportunity afforded for state surveys has been improved by quite a number of states, ujiile the country will eventually be covered with a tri- angulation netwhioh will compare fa\forably with any in Europe. Since the extension to include interior work, the survey has been known as the Coast and Geodetic Survey. It is under •t'he, Treasury Department. The superintendents, and times of their appointments, have been,f.B. Sas- sier, 1807; A. D.Bache, 1343, Benjamin Pierce, 1367; C.P.Patterson, 1374; J.B. Hilgard ,1331;P.M. Thorn, 1338, T.C.«endanhall,1389;(I.W.Duffield,1894; H. 3. Pritohett,1397; O.H.Tittmami,1900. The yearly reports contain inich Val- uable material .especially in the appendices. The survey of the siorthern and Northwestern Lakes was commenced in 1341, under the (Tar Department; better instruments and methods were introduced in 1351, and the character of the work was gradually iaproved to 1370, when the survey passed under the charge of Gen.O.\f..Comstock of the Corps of En- gineers. From that date to the close in- 1881 a continuous chain of tri- angulation, depending upon 3 carefully measured bases, was extended from St. Ignaoe Island, on the north shore of Lake Superior, to Parkersburg in South- ern Illinois, a distance in latitude of 10°, and from Duluth,Minn. .via.Chi- cago.to the east end of Lake Ontario, a distance along its a:xis of 1,300 miles, or in longitude of 13°. Some very excellent base-line work has been done and the triangulation has been carefully executed. See, Primary Triangula^tion J.S.Lake Survey, 133£, by Gen.C.e.Comstock;or see the year- ly reports of the Chief of Engineers. Many of the states- are now engaged in geodetic surveys. Jiassachus- setts took the iead,ander_Borden, in 1331. CHAPTER 11. TRIASQULATIOS,RECO!IBOISSANCB, SIGHALS. 5 FRIlSARi SS0OHDARX,TSRTIAR!f,TRIASGULATIOH. Wlien a triangulation is to be extended over a large tract of country, or between two or more dis- tantplmsTs^stem of primary triangles is employed;which is character- iZl S the maximum development of which the topography will admit. This nrievel or slightly undulating country, will allow of triangle sides of only 15 to S mUes!on account of the height of signal,and of cbserving stand,required to overcome the earth's carvature;while in "o^^t^^g"^^ S?5Slrl^?LSln5-Bit?aSSfa?i^|tMrS?^g^^ 1 • 100 000 the range being from about 1 : 60,000 to i . AOD. Then in the right triangle AOD, ? 2 2 k'= , )N_ (AC + H) k + R ,or AO.in miles * j^.nearly af^ •.• ADB= mAOD = 2mADC,and the angles are small, .-. AB £r,&0,and BC.in miles, or _h ^ 5£S0 AC-AB= J! (1 -2in) ; b -^ =0.38^530 2 R k"- = 1.743 h ^'^' irhere k is in mileS' and h is in feet. 'o I.e. .tfeg sq uare of the distance jjr miles ia about 1 3/4 times the re- qiured elevation in feet :- a convenient rule easily remenbered. For k in kilometers and h in meters, (1) reduces to k^ = 14.807 h (i£) T^e line of sight should not pass nearer the surface than 10 feet at the tangent point, on account of the lack of transparency and danger of lateral refraction, due to the disturbed lower air. Ex.1. Tirp stations of the O.S'. Lake 3arvey, Buchanan on the north side of Lake Superior. and BruS River on the south, are 10 and 19 feet above lake level. respeotWely.and 16 miles apart. A signal" 35 feet high was used at Bruit. Bow high should. the instrament and observing stand be elevated at Buchan- an, in order to see the upper SO feet of the signal at Brul6? 19 + 35 - 20 = 34; 34 - 10 = 24, the ava ilable height at Brule. Placing h = 24 in (1), k = v'1.743 » 24 = 6.5 miles, the distance from BrulS to the tangent point. IS - 6.5 = 9.5 miles, the distance from the tan- gent point to Buchanan. h' (9.5)2 "ijH?' = 52 feet. 52 + 10 = &2,the reonired height above lake level, or 52 feet above the ground. 8. HINTS IN SSLECTING STATIONS. Choose the highest elevation? even if S GEODSST. K9, Pig. 2. at greater first, cost on aocoant of inaccessibility. They Bill thsa the better conniaDci de» ground, if at ansj time it becomes aeoessary to extend the Bork beyond its original lii7iits;-,»hile high lines of sight iseet lass atmospheric distarbance. Use as long lines as the topography of the country, and the visibility of the signals, trill admit of in order to increase the accuracy. Avoid Ion lines and lines passing over cities, furnaces, etc. Porn triangles nhich shall be as nearly eqnilateral as may be; the as- nal limits for an angle are from 30° Jro 120°, but Capt.Boutelle no« recom- mends for C.and G. Survey practioeCBeport ]S85,App. 10) an extension of from 10° to 15° each iray in Quadrilaterals or other well checked 'systemsi of primary tria ngnlation irhen necessary. The nearer an angle to 90° the less does a change in its value affect its sine, while the nearer to 0° or 180°, the greater in an increasing ra- tio does a change in its value affect its sine. Hence a triangle side will be least affected by angle errors, when the anglesoo which it depends are near 90° The nearest approach to this, when two sides of a triangle are required in terms of the third, will be 60° for each' angle ,as given above. If, how- ever, one side is not common to any other triangle- as when advancing by a single string of triangles- an error in its length will not be transmit- ted into tie chain,, so that a small opposite angle will not be^objeotion- able as when both sides are required with equal accuracy. )Ihen a point is to be located by cuts from two or more known stations the lines should intersect as nearly at right angles as may be. In finally locating stations, make certain that 'those intended to be iD'teFvisible really are so, even at the expense of time and patience in waiting for clearing weather ; otherwise the observing party will suffer vexatious and ekpeasive delays. Select stations so that permanent station-marks can be placed and pro- tected,or so that accurate references can be had to permanent objects. Advance by quadrilaterals, when the greatest accuracy is desired. Locate secondary and tertiary stations so as to command a sweep of th$ area to be surveyed, in order to readily locate, by intersections, points- for the topographic and bydrographic parties, 9. BASE IiIHBS. A base line site should be selected with reference to securing suitable ground for measurement and a convenient expansion, by wpll shaped triangles or quadrilaterals. to reach a side of the main tri- angulation. The line should be free from obstructions, and quite smooth for a widtb of at least 12 feet; longitudinal slopes up to 3° to 5° are admitted without serious inconvenience. even when making the most accurate meas' aresents; the ends need not be intervisible from the ground, if they can be made intervisible by signals and observing stands of moderate eleva - tion. The measurements can be made along two straight segments, not dif- fering widely in direction, if better ground will thus be secured. Harrow ravines can be crossed by bridges or trestlework with complete success; while a wide one, or a bog or similar obstruction fo direct measuremeat , can be passed by triangulation without very serious decrease of accuracy. Sabsidary bases which- are to be measured with a long steel tape can be located on rougher ground if necessary. The selection of the system of triangles by which the side of a main triangle can be computed from the base. with the greatest accuracy for the expSaditure, requires considerable skill . Auxil iary stations will be re- quired in the expansion;working down from a side and locating the auxil- iaries and bese- line to correspond in a level country, or up from a base- line to the main side, modified to adapt it to expansion if necessary, in case of rough country. la case several sites are available, the cost of preparation and of measurement, and the cost of the connecting triangulation. should be esti- mated for each; this when compared with the relative accuracy of the tri- angle side which each can furnish, will allow of selecting the one most Ex. 1. The Buffalo, base of the 0.3. L. Survey. measured near Buffalo , BEC0NN0I33A1ICS. S.T. ,in 1375, is slio»n in Pig. 3. together (ritli the oonaeotioa to. and a por- tion of. the main trian- galatioD. The gradual enlargement from the base to a side of the main triangala- tion.and the different triangles rhicb may be nsed in finding the length of any side, as Grand River-Westfield.from the base may be noted. The Edisto base of .the C.Survey, shown in Hg.5,%1.1. on the other hand consists of. the side of a main primary triangle; the other sides being- short because the country is leyel and hearily timbered. 10. BBC»NHOISSMQE. PRIMARY THIiSGULA'TlON. A general reconnoissaace should precede the selection of stations. in order to become sufficient- ly familiar nith the topography to be able to recognize the most promi- nent features and ele7ations.as seen from different points of yiew^and in order to determine the general scheme of triangulation.and the gen- eral routes best suited to the ground. for aid in conducting the detail- ed reconnoissance. Unless the'sarfaoe is leyel and unbroken .points irill be found which from their position or eleyation.nill offer such advantages that they probably must be used for stations. Starting from these, others must lie within prescribed areas, in order to fulfill the required geometric conditions. and make use of the longest feasible sides. ^om each of these probable station points. sights should be taken to the others if visible. and also to such points in the prescribed areas as uill possibly serve for stations. Other available points can be occupied. and the process repeated.if nec- essary. Should a point be occupied irhioh has not been cat from at least two other stations.sights must be taken upon at least three knoTrn points. »hen its position can be determined .by §12. Magnetic bearings often aid in orientation on arriving at a new station* and in identifying objects already located. by giving approximate direc - tions; while they sometimes aid in plotting when insufficient angles have been taken. A hasty outline profile sketch of the ground in the vicinity of each ob- ject sighted sill aid very materially in identification from surrounding stations. while if the estimated distance in miles.is written near the point.and the circle reading is written above on a vertical through it, see Pig. 4. very clear and concise notes- will result. The obstructed arcs at a ^ •, ,^ ,^ station should be noted; as also the cause. and whether they can be removed by cutting, or by signal elevation. Should the location be likely to prove difficult; vertical angles should be ta- ken to aid in deciding upon the inter visibilitB of signals by giving differences of elevation, A plat of this preliminary triangulation should be kept op by angles, starting from a known or assumed side;or by computed triangle sides.xf greater accuracy is desired. Then wprking from probable station points., or from stations already located, the possible point in a given area is picked out which will best fulfill the conditions imposed.as to length of line. intervislbility. etc. In the same manner as many new ones are chosen from the plot as desired. Without experienoe.it is quite difficult on reaching an elevated point, to orient one's- self and be able to identify signals and topographic fea- tures- at distances of 40 to 50 miles.even under the most favorable condi- tions. iJhen.as is often the oase.the features are not prominent. and the air is thick with haze and smoke for days at a timeythe skill and pa - tience of the experienced are fully taxed, with wooded elevations the observations must be usually taken from the top of a tree.or if none can 3 < •■ ^^ ■N.> ^ "* ^ic^.l. 10 fflODBSV. 611, Pig. 4, be foana of sufficient height, from the top of a ladder formed by splicing seyeral together and supporting them by gays. High elevations with the-sammita free from timber afford the best sta - tion sites. ifooded summi'ts require sight-rlines to be cut throagh. These for pole signals should be about 100 feet ride and they should be extend- ed back of the station far enough so that the signal will not be seen a- gainst near npods. As the summits broaden, or the timber becomes valuable, elevated signals and observing stands should be considered before clearing the lines, al- though they generally should not be adopted unless a considerable saving lill result. Parallel looded ridges may present much difficulty, if so near together that the triangle sides must reach over an intermediate ridge instead of spanning an intermediate valley. The direction across the ridge to an Vnvisible station can be found from the plat, or from %14;!rhen the re; quired signal elevation can be found ,from the vertical angle, or from ca,rsfally taken aneroid barometer readings; but if tiro or more ridges intervene. actual tests, from ladder. tops <, or an examination of the entire line Kill be necessary. In level country, an elevation of 70 feet for signal and observing stand irill allor of SO-mile sides. If *ooded, these had best be used in a chain , of nearly equilateral triangles having all- the lines out through; but it clear, as on prairie, quadrilaterals with diagonals of 31 miles and sides of about 15, trill add only one more station in 30 miles of progress. which vill be more than compensated for by the increased precision attained. If the level ground be cultivated and contain patches of valuable tim- ber, the difficulties vjll be so much increased, even if the ground be rollr ing,that the greatest care and skill irlll be required to avoid Insuper- able obstacles. Sometimes chains of secondary triangles along the irar ter courses, have proved effective. Full notes and sketches should be taken of the points most important for the subsequent viork. Among these are the means Of access; the timber ■wtioh can be found at the site for the signal; the roads which have to be opened by the angle party in occupying the station; the places nearby irhere board can be had; etc. The efficiency and economy of the survey HilL depend very materially upon the skill, good judgment and experience of the person irho conducts the reconnoissamce. 11. SSOOHDABr AND TESTIABr THIAHGDM'EIOH. Starting irith the long primary sideS' as bases, points of the first order are taken, uhich Hill shorten the triangle sides and command the area to be surveyed. Prom these shorter sides, points of the second order a.T% taken so that they iKjll command every prominent object visible. " From the short sides thus obtained. tertiary points are located by eats from at least Z preferably 3, stations-. ' These points should include aS' many prominent objects, usually from 1 to 3 miles apart, as may be needed by the topographer in tying up his. wprk, or by the hydrographer in taking angles to locate soundings. etc. ; such as charok ssires.,oupolas,chiBneys, flags in prominent trees, large white cross- es' or triangles' painted upon rocky cliffs, etc. Well-shaped triangles are not so important as the securing of a sufficient number of convenient points- for the topographer, since the errors intro- duced do not accumulate over large areas. beiag checked by the primary sys-' tern. If the latter is omitted, better shaped secondary triangles should of course be employed. , Bx. 1. Fig. 5 shoirS' a portion of the primary and secondary triangalation near the gdisto base of the and S. Survey.South Carolina, on a scale 'of 1 : 400,000, taken from the Report for laes.App.lO. The country is flat and ifpoded.no elevations of 80 feet being available. The use for seooo- dary sides of the lines cleared for primary ones may be noted. In: the same App. may be found a sketch of the secondary triasgalatioa of Boston Bay, an open country uith suitable elevations-. It *1"f*^ «,,„ B.pemra problbm, 12.lf-P0I»T PROBLEM. To deter- mine the positioQ of a point.when only angles at the point have ■ been observed between knoioi sta- tions. Lay off the angles on trac- ing cloth in order around a point: plaoe the cloth on the plat and B07e it until each line shall pass- throng the station to ihlch it belongs; irheii the 7ertex can be p^et ad through. Tiro angles iiill locate a point, giving the 3-point prob- lem, except nhen the point lies on or near a circle passing through the three stations on irhich the sights are taken; 3 or more angles are better. forming a check. A 3-armed pro- tractor is often used in place of the tracing cloth; also a sheet of pa per. by cutting out a narroir strip along each line near the portion to be used. (Then a more accurate solution is- desired than' can be had from a careful plat on a large scale, a nu - merical one is used. In irig.6 .let S be the re - quired point at which the angles P .P'^P*. — ..., hare been observed upon the knovn stations. F.G.H.. B is also knovD.it being the angle between knoirn stations. In the triangles. SF6;S14. Fig. 8, + 18.486. 2. 56807 8.56307-'' 1.21711 17.486 - 1.84269 9.14292 0=82° 05.3,oot A 28 22.1, cot 0.38561 .4 by (S) C 59 43.8 Q a 7150.3 3.85432 P+A=140°37.3,3iQ - -9.80239 P = 118''15.2,sin n = 5149.9 3.65671 9.94491 3.71130 b - 3050.7 C=59°43. 2 , SIQ - 3.48440 - 9.93630 3.43D70 P' =53° 31'. 8, sin -9.93036 n' 3039. . a . - - 3. 43984 Computing a' by the first eqaas. of (a) the same valae is found as abo^e. 13. WO-POIHT PROBLEM. If tifo ualino.fji stations, C and D, Pig.?, see each other, and also tuo kno»ji stations, A and B, their positions can bs deter - jnined by measuring the angles 4CB ,BCD,CDA,ADB,3S follows: Draw the line C^iy of convenient lengtn on tracing cloth and at C and !f , lay off bae measured angles; the intersection of the tsja lineS' ffliioh pass throa.gh A will determine its position on the cloth, and similarly for B; join A and B'j place a:^ the cloth on the plat so that A', uill coincide sith station A and B'. will fall on the line AB of the map, produced if necessary; prick through the points B'.C. and D'. Then through E dra» //'s to B'C and B'.D'.; their intersections with AC and AD', uill determine C and D on the map. If more accuracy is desired; assume' CD as unity and compute AC and AD in the triangle ADD. and BC and BO in the triangle BCD. Having two sides and the included angle in ACB.AB can be found (formula 33): the ratio of the true value to the com- puted one will be the ratio which the other sides bear to their computed values. Ex. 1. The following angles were observed at Giles and Blm of the CO. S'kan^eateles Oake Survey in 1892 upon the known sta - tions Eaight and Olmstead. Fiq,n. Ha-j^t 02'. 17" 05 03 01 87 04 29 EXm Haight - Qiles - Blm = 50° 01m. - Giles - Elm = 35 Giles - Elm - 01m. = 38 Giles - Elm - Haight = 50. Haight-Olmstead = 18944 feet. Ci.i^^ For fuller treatment of the N-aad tTO-POint prob- pi^,^ lems.see Zeit,- fur 7ermes,1333,P.140. 14. DISECTIOH OP INVISIBLE STATIONS. It enough an- gles have been- taken so that the stations can be platted by methods al - ready given, the direction of the line joining any two can be taken di - reotly from the plat with a protractor. Or, starting from some known side, the sides of the preliminary triangles can be computed from the ob- sprved aneles:when by assuming a meridian the distance in latitude and in longulde of each point from an initial one can be computed as in an ordinary land survey. The tangent of the azimuth of the line joining any two points can then be found by dividing the difference in longitude by that in latitude. The line can then be cleared from either end if obstructed by timber, or the height of signal for intervisibility can be determiaed if the obstruction is an intervening ridge. Sq.5.] POLE SIOTALS-. 13 Por an examplg in diffioalt ooantry In aoptUem Alabama, see IT.S'.C. S S.S' 8eport,13S5,4pp. 10. Ififo stations and D each see wo points A and 3, Fig. 7 %13. ttie di- rection to tciu from one to the other can then be fonnd as folloira: At k measare BAD and DAC.and at B.OBO and CBA. Compate AD in the triangle ABD and AO in the triangle ABC. calling AS unity; then in AOD tup sides- and the incladed angle are -knoirn from »tiioh thS angles at C and D can be foand by formula 203. Or . the dipections can be found by platting. 15. OOTPm. When accurate angles are required a ligi}t tranr sit with a good telescope is most convenient. The needle 'itill givre bear- ings, ifhile by addin.g a level to the telescope tube and a gradienter soren" or good yertioal circle, elevation angles can be measured ifith sufficient accuracy for dstermining intervisibility. An aneroid barometer is also convenient for determining differences of elevatio.n. For distances over 25 mileS'.a reconnoitering glass with stand will be found desirable on ac- count of the larger telescope. If care is taken in setting up to place the tripod head level. the small horizontal circle will give angles quite accurately. In a wooded country wbere angles have to be measured from tree tops, a sextant will be necessary; also a telescope or field glass for identifying the stations, and a set of spurs or creepers , for climbing. An azimuth or pocket compass is convenient; also the best available map of the region. To these should be added some IDO feet of about 3/3 inch manilla rope . a ball of tffine.an axe, and material for different colored flags to be spread out upon trees or other objects for temporary signals. An assist- ant. who ISi quick and handy at all kinds of work and wdo is used to climb- ing.and a horse and covered wagon, will complete the outfit, Jfuoh of the traveling will necessarily be on' foot or possibly on horseback.if the country is hilly or wpoded. if auay from all supplies, a cook and the usual camp outfit sill be necessary; irtile for primary triangulation,in rough country uith good railroad faoilitiesflike much of Heif England.it may be more convenient to travel the long distances between stations- by rail, hiring a horse when use can be made of one. 13. SIGNAIiS'. After the exact station points have been located, the sig- nals ttjiioh are to be erected over them, to give definite points for sight ing in measorina the angles should fulfill the follosfing conditions: $hey should be cons-picuous.so as to be readily seen and distinguished from surrounding objects; they should have a irell defined central line or point upon wbich to fix the cross-hairs; thay should have Httle or no phase , i.e.. this line or point should not change in apparent position with the direction of the illumination by direct sunlight; they should be firm in position anless of the class- which require an attendant; they should Ee oheap.or light and portable; irhile often it is- convenient if »hen in place they will allow an instrument to be set up over the station point, with tnese gemersl requirements in mind, the. relative advantages offered by the different signals- to be described uill be more readily appreciated. 17. POtiS SIGHACiS. #lien height is not required for inter visibility.one of the most common forms of signal consists of a vertical pole set in or on the ground. and supported by braces or nire guys; or of a pyramid or tripod surmounted by a pole. On sharp mountain peaks, wjiere only small, stunted timber can be found, the < rectangular pyramid.Pig.e, is- convenient. 4 signal 1 'j'"^'"^^*- with height of 'apex of from 12 to 13 feet and f^ svi^.-.-v. legs from 8 to 5 inches at Jie top. can be erected BL ^^'*' and a center pole 8 to 12 feet long inserted by mSk 3 men. without tackle. By inclosing the top with J^S^ boards, cloth or slats made from small poles, vis /,? '. I<\ ibility can be given';wtiile the apex and pole re- JLJa^JCmLm main for accurate bisection. The pole can be in- / ilJT \ / \ creased to any desired diameter by nailing on ^TJ^^^lT "^ slats- or poles after erection; while the signal can /-JS. \/\ _ be anchored to the rock, by wiring the legs to an- X ' /''l!^=^^if» fliiDjrJx)ltS.ar by wire guys extending from the top -^fe" / =--- — 14 GEODSSY. t|l«.Pig.ll. of the pole. On flatter peaks, more height mast be given for visibility. rendering the trlpoa signal, ?ig. 11, more oonvenleBt, By bolting all foar pieces to- gether on the groond.irlth a 1 to 1 1/.4 inch bolt ,as shown in Pig, ID or better uith the head raised 6 feet on a bent or staging. 5 men can raise a 25 to 35 foot signal of round timber, each piece being 5 or 8 inches in diameter at the top.iTith no special ontfit except about 30 feet of rope. Pits are dug.or stoneS piled up to prerent the feet a and b from slipping; the head c i» then lifted and pushed to tx>sition by the third leg ifhen the pole is made verti- , oal by pulling do/rji t^e large end *ith a rope; it is secured by spiking braoes' to the tripod If' the angles at the station are to be meas- ared uith the signal in place. the legs should be so placed as not to obstruct the lines- of sight to the other stations. They should ex- tend a couple of feet into the ground;or if on- rock, be securelr tied to anchor bolt3 by rtre rope.or notched and horizontal cross, pieces at- tached and loaded uith stone. Hire guys from the top of the pole may also be desirable fi tin cone or barrel of larger diaideter than the pole is often placed at the to p. especial- ly irhen the tripod head irill not be seen against the sky. The pole should not be more than 6 to 8 inches at the tripod head.even for a large signal.on account of the weight in ereotion;it can afterirardS' be inoreased.or the pole stcdight^ned . by nailing on light slats. Or.uhen lumber is available. a square bolt of E- inch plank in place of the pole will give diameter without increased )reight;one or more slats along the center of each side irill make it more nearly cylindrical. A very convenient ani3 portable signal for tertiary irprk can be made by supporting a pole on a tripod having a light cast iron head and about 10- ft.legs. By holding the pole in position by irire guys, a signal 15 to 30 feet high can be made very stable while there is room enou4h underneath to sat up an instrument. Any portion of the pole can be enlarged to any desired diameter by light slats. 13. DltMEDSa ftHD SEIfflt. The diameter of pole for short lines, may be large enough to subtend an angle as seen by the observer of 4 or 6 se - cond3;but as the distance amd the power of the telescope increase the angle should diminish, according to Goast Survey practice, iowp to one second for about 15 miles, and not fall below; this valuje for greater dis- tances ( see also §19K Biameter to subtend one second at, 1 mile =; 0.307 inch. 40 miles - 12.3 inches 10 ° ^ 3.100 •• 60 » = IS. 4 » 30 •: * 6. 130 80 " 34.8 » Increased diameter beyond that necessary for visibility gives increas- ed range to the cross-hairs, in bisection, and introduces the uncertain' ele- ment of phaa.9 "ith cylinflrical signals which do not show, against the S'ky. The height of signal in feet should be aboat ohe-half the distance in miles, plus 10. Less height may answer for long lines. or for signals on sharp peaks with a sky back ground. but height adds to visibility without diminishing accuracy.and with only the increased cost of construction. & signal to be seen against the sky should be painted black or wound with black cloth .one to be seen against the ground should be painted white or wound with white cloth; unless two colors are needed on; the same signal for ready identification from sarroanding objests.when the pole. S(i,6.1 gLSVATSD SIGH&IS. 15 or pals and tripod, can be painted in alternate rings of black and irdite, or red and ibite.each ring being several feet wjlde. 19, SIGNALS. iriTaOin! PHASe.- Various signals have been devised to avoid phase or the effect prodaeed by the unequal illnmiaBtioi by di- rect sanlight of the portion of the' signal facing the observer. irhereby the apparent and real centers' do not coincide^ ' One devised by Be^sel for the Prassian triangilation in 1331, and used on the 0.'.S'.Lake Sur? vey, consists' of a board in place.or in front of. the pole with its face X to the line of sight. -On the latter sarvey a width r^a, giv - en of about 4 seconds- as' seen by the observer, yet good angles rere, ob- tained. The station ims't be visited and the board changed each time the observing party move to a ne?r -station. Another designed in 1381, and used on the Mississippi Biver Survey for distances' of from 5 to 12 mileS'. gave excellent results. It consists of a horizontsyl board 6 inches in' diameter, to the circumference of irhich are attached 4 stiff vertical ivires^90° apart. each 5 feet long. These !7ireS' are held in position by a wire ring at the top and another one-third the distance from the top; each joint beinS well soldered. Tio ooposite »ires are connected for the upper and loner thirds by a 'thite cloth, and the oth- er two for the central third by a black cloth; 4 gay -fires are attached at the central ring^and the board rests on a tripod or otl^er support. aO. BIiE'/ATED SIGNAEja- AND 0BSBB7I11G •STANDS. »hen the signal and in- strument at the station- require elevating.and no existing structure can be made use of,a suitable one must be erected. -The standard tripod and scaffold adopted for C and 6. Survey irprk. for heights of floor from 32 to 98 feet, increasing by multiples of 16, are shosjr in Pig. 12. The scaf- fold is removed from the tripod in elevation for clearness-; their relative positions can be seen from the plan. For full details see Capt. Boutelle'S excellent paper in' Heport ,lB38,App. ID. See also, App. 9. page 158, and Pri. Iri.O.S'. L. Survey, page 318. The tripod, x^ich supports- the instra;- ment when observing and the pole or other signal «^eu- observ- ed upon, starts vfjth a firm cap; the posts- are 6 by 8 inches-; thsy are scarf-spliced ffith a 3-foot lap. held by 6 5/8-inoh bolts and 4 5-inoh boat spike, at points 33 ft. apart starting from the top irith 3S-ft, sticks; batter 1 in 8; and bracej by joists from 2 V<\ 3 to 3 by 3 ins. spiked irith 6-inch boat spikes. The observing scaffold. -(fhioh is placed outside of but not in contact ijitii the tripod. starts -ffith a floor 12 ft. square about 4 feet below the tripod head; the posts are 6 by 6 ins.; in sections of the same length and spliced in the same manner as for the tripod, using half-inch bolts; batter 1 in 6 measured diagonally; braces from 3 by 3 to 4 by 4 ins, in 16 ft. tiers. The posts above the floor are connected by a railing;»hiie the flight of stairs connects the landing on the top of one set of horizontal braces nith that on the top of the next. The sjiort central posts starting on the ground in Fig. 12 are only used for tall scaffolds. The posts for both tripod and scaffold rest on wooden shoes 12 by 15- inches-. They are all placed on the same level, about 3 feet below the ' ■■•:v-:.^" <-'i<: I / 'rK // PX«>.»k. 5u>^>\'^ ^■*V-yoi ? f GSODSSY. [5x0, Pig. 13. siaiioTi point; and at the proper distance apart and from the center, by plambing do»(i from a templet placed on the ground. To erect a structnre of 3 sections : a derrick boom about 30 feet long and a, inches in diameter is set u.p and held by guy ropes, advantage being taken of a tree if conyenient in erecting it, the lower lengths of the tripod posts are then lifted npright,one by one, and held by gays- .fith the lowpr ends- in position, a workman ascends each post by meanS' of cleatsi fastened to it, and the tops- are sprung to relative positions and nailed to a templet, the templet is then shifted until a plumb hung fron it3 center will fall over the station-point; uben the bracing is 'spiked on and a floor laid on the upper horizontal joists. The pulley block is. shifted to the top of a post and the lo«f;er end of the boom draifn up to the floor, it being kept upright by paying oat the guys attached to the top; the next lengths of posts- are dra/ra up and the splices bolted; the tops pit in place and the bracing attached as before. The derrick-* is- lowered and the loirer tiro sections of the -scaffold erected and braced as above; a floor is laid over the horizontal braces of tripod and soaf - fold ;the derrick is- draup' up and the upper section of each put in place and braced. About 12 days will be necessary, Kith a Mftn.-xot-'Toipc. ■i^- ,.is-. 18 ssoDEsr. Ggs.Pia.ie. pasteboard or otDer screen witb a smaller opening should be attached to the second ring. For longer linea larger mirrors are used. Thus on the IT.S. Lake SurTe; for the longest lines a common mirror 9 by 12 inches fas set up and light thrown through a circular hole in a Kooden screen some SO ft. distant in the direction of the obserring station, this having a diameter of from S to 10 ins. on sides of 90 to 100 miles. On the longest line ever observed, Mts.Staasta-Lola in northern Gal., 192 miles, a helio 12 ins. square Tras used. vfilson. Topographic Surveying. gives X = .046 a (6) for the length of the side of the mirror in inches, nhere the distance d is in miles. and d>10. Too mach light gives by irradation a diameter too large for accarate M--^ section and increases, the unsteadiness; an opening suited to the distance or one which will subtend from one-fourth to one-fifth of a second ,will give in quiet air a small bright disk easy to bisect. 4n intelligent and very faithful person should be picked out for the heliotroper; othernjse delay and vexation irill result. If he is to oc- cupy the station a long time he can usually be picked up in the locality irith economy, if for only a short time it may be more economical to have one Kho is familiar enough uith the work and irlth instrnments to go to neiT stations and establish himself trlthout assistance, vhen directed by the observing party. 22. HIiHiT S'IGNADS'. [/amps ffjth 10-in. reflectors for short lines and the Drnnmond light for long. ones upre used on the B.nglish Ordnanoe Sur- vey in the last aentury:iihile night signals have been extensively used in the recent prolongations of the Mbuvelle miridienne de France by U. Psr- rier. and Argand lamps and heliotropes are exclusively used in India. The electric light, in the focus of a reflector 30 inches in diameter and 24 inches focal length, proved very successful recently on a line of 1S8 miles across' the Mediterranean ifhere on aooount of fog and mist a 12-inoh heliotrope had failed to once shoii. during a three months' trial. Some recent experiments' made irith the magnesium .light' indicate that it is sufficiently poijerful for long liue3;»bile. unlike the Drnmmond or sleet- ^vc light.it is exceedingly portable( the instrument used pteighing only "5 IbsJ and can be operated by an ordinary heliotroper. The apparatus consists of an 8-inoh reflector, a small lamp.a clock work, and a reel of magnesium tape »hich is fed by the clock to the lamp and burned in the foous of the reflector board screen iras used to redace the diameter on all but hazy nights on a line of 60 milesi The tape ^as. burned intermit- tently by time table to save expense; it costing about 2 1/3 centS' per minute for a steady continuous' light. Two of M. Perrier'S' lamps- ifgre also used. See Pig. 16. Bach consists' of a box con - taining a flat wick petrole- um lamp in the focus of an 8 inch lens of 24 inches' fo- cal length. The emergent rays snbtend an angle of about V. ujth the magnesium iras about as- 2 to 5. For aoouratfe bisection a paste - Fig. |6. — CollimatQur oplique. The intensitij of light as compared 5. It made a very pretty mark to point upon on clear nights.but at a distance of 43 miles it would often be scarcely visible m the telescope, and would not allow of iUnminatin/ the cross -hairs.when the magnesium light was clearly visible. A student lamp was also tried; and with an 8-inoh reflector it was visible in the telescope at 31 miles when the outline of the mountain was invisible at sunset. The accuracy in these experiments "^.S'.C.S G.S', Report. 1880. 4pp 8. proved to be equal or greater than '9 ■ , S'TATIOH RSFSRfiSCB. ,q J^L^i^"^ i '5^^^ '"^ ^^"^ ^""^ ^°°'^ obserying in fa7orabls xeather ended from about one boar after saaset to from 10 o'olook to lidnight llimators and reflectors, with kerosene lamps, were both saooessfully a_on the H.y.State survey for distances up to about 50 miles. The ■ i ?*^!f ;?^^ ^^" ^'■''^ ^="^ *"? OEOSS hairs illaminated from behind, lag light lines in the dark field. he jfhitecess and intensity of the aoetyline Ixght.and the simplicity the portable lamp,should place it in the first rank for sight signals. S. ST4TI0N HEPBRSilCS. In referencing a station, the object shoold be render the recovery of the locality and of the exact station point easy and certain as possible. at any time and by any one unfamiliar h the country but familiar »:ith the kind of work. The station point tasually marked by an underground and by a surface mark., .The under- und mark should be placed below frost. and plo».or some 3 oc 4 feet Off the surface. It may consist of any material which is durable, for- V to the looality.and capable of receiving and retaining an exact cen- mark. ugs and bottles, cat s>tane blooks.and holloir cones of stoneirare are a- ^the most common, (the stone blook,holding a copper bolt, and sur » itdea by masonry is much used at the ends of base -lines, ; is divided by the nnnber oi repetitions for the value of the angle. In measuring vertical angles a level on the. side of the loirer teles - cope comes u.p in position, not showji in Fig. 17, to serve for the refer - ence horizon when the circle is vertical. At the same time the English brought forward the celebrated Samsden th-eodolite, partially described in SS.which in its essential principles is the same as the modern theodolite and does not need separate desorip- 'tlon Bq. 6) THS ASTROSOIilCAL TELESCOPE. 21 The different parts of an instrumeat niil be taken ap in detail, begin - ning xith the telescope. £5. NORMA!/ WSION. The eye is an optical instrament, consisting es - sentially of a series of transparent refracting media bonnded by carved surfaces, forming a lens, and a delicate netxork of nerre fibers, spreading out from the optic nerve, forming the retina. A pencil of light entering the eye is refracted by the lens and broaghi; to a focas upon the retina, and the impression is carried to the brain along the optic nerve. The normal eye at rest is sapposed to be adjusted for parallel rays , the curvature of the lens and its distance from the retina sill increase itith the nearness of the object up to the limit -if distinct vision.Thioh is some S to 10 inchesj the pupil or aperture for the admission of light is also adjustable. ?he distance from the center of the lens to the re- tina is about 0.6 inch. ^r, a * oi Bith this ratio of distances of retina and object from lens tO.B to a) the image will .be only 0.6/8 = .075 times as large as the projected ob- ""^The angular magnitude for 1** in the projected object at the distance of 8 inches ,irhereFi= the millionth part of a meter, =0.000,0394 inolies _ 0.000.0394 * yr S sin 1' The minimm angla bet;?een ti»o bright points or lines upon a dark ground.or the reverse , . vhich the eye can distingiish Tithout running them together is found to be about 60*. This .vould give the distance betireen the images, = 60 X .075 " 4.5y The surface of the retina is made up of mi- nute papilla or nerve elements called saiiS and cones from 2»» to 6h in diameter, nith an average of 4.5*';' shoning no power to distinguish impressions on parts of a papil- 1ns. A single dark line upon a bright ground can be distinguished, it is said.when the visual angle is only l/50th as large as the above {-iniage 0.09H>. AccoEiUDg to Pfr. Porster's investigateons as given io Jordan's Bandbach der V«rffiess.,7ol.II,p. 147,the mioimam distance b,bet- veen a hair and scratch, nhich can be distinguished in bisecting .a division mark upon a bright scale, as nith the cross hairs of a micrometer microscope, Pig. 19, is 2.5f measured upon the retina. ITith this width of line the probable error of the bisection, with a power of 25, was found to be 0.25*'measured upon the retina. This width referred to the object and unaided vis- ion, would correspond to b • 2.5/. 075 =, 34'^or a visual angle of 34'';while the probable error of bisection would be one-tenth as great. A poiter of 34 would thus give a probable error of 0. 1»* in bisecting a division. If b be increased 16 fold,or so as to cover 8 papilla or nerve elements a power of 85 is necessary for a probable error of O.l**- in bisection ; and if widened to cover 15, a power of 150 is necessary. 26. THE ASTROMOUICAL TSLESOOPS. This in its simplest form consists of two biconvex lenses fixed -in a tube; the eyepiece and the object glass. Its advantages over the unaided eye in .accurately sighting an instrument upon a point ,are; (a) increased light; (b) magnifying po.ter; and (c)the use of cross hairs. The folloifiBi are from Geometric Optics: A lens is "a 'portion of a refracting medium bounded by two surfaces of revolution having a common axis; this axis is called the axis of the lens " The surfaces of revolution are usually spherical or plane; if they "do not intersect, the lens is supposed to be bounded by a cylinder in addition having the same axis. The thickness is the distance be- tween the boundini surfaces measured on the axis. The optical center is a point of the~axis,usially within the leas, through which if any ray of light pass, the direction after passing through the lens will be par- allel to^its direction before, a slight.pffset takingjlacfipr obUque ) i Pi,. IS. i,2 GEODESY. C527,Pig.21, rays oq aoooaat of the refraction towards tlie normal on entering tlie lens. For sphepioal sarfaoes this point is found by drawing any tuo parallel radii, joining the points where each cats its oirn sapfaoe, and noting the intersection of this line with the axis. The ratio of the dis-T tanoes of the centers of cnrratare from the optical center eqaals the ratio of the radii. Hheij one surface i^ plane, the optical center is found at the other surface. The principal focal length of. the lens, f, is found fron. Fiij.ao. (7) Fiq.41. where r and r' are the radii, and fi the index of refraction. The fundaoental equation connecting conjugate focii is, ■f '■? +F (8) wSere f is the distance of the object and f that of the image. 37. MAGNIKING POiTBB. In Pig.21,let be the object (glass and a', the eyepiece. The rays of light from the arrow head,& will ~be brought to a focus at A' where the ray through the optical center meets the focal plane, and those from G at C. .these rays preserving their direction be- yond the lens but saffering^^slight offset as indicated in §26. Join A' and C with the optical center of the eyepiece. All' the rays of light comin;g from A and C which pass through the telescope will e - merge in pencils parallel with, or slightly di7erging from.these two « directions A'.C.C.O'., if ad - justed for distinct vision for a normal eye. Without the telescope, the angular magni- tude of the object with the eye at would be P. Tlith the telescope, the angular magnitude iS"*. Draw ffi = f.,the focal length of the objective; erect thexKJ - A'.C/2;take PS fi .the focal length of the eyepiece; erect the x KL = HJ; join J and [• with P, giving HPJ~ = P/2,an,d ap« =«yg. Extending PL to U to refer both images to the same distance, the appar- ent magnitudes will be as BH to EJ. But HM : HJ - PH. : PK = fifj ,or I.e. ,t^ ija|ai|yjjLg power equals the focal length of the object glass over that of the eyepiece . ' — — ~ — — " — * ■ Also, HU : HJ ■! tan °rhich may be condensed to enter it is D. The quantities of light, for same distance from object, will vary as the squares of these diameters, or allowing for the loss diie to absorption and reflection of the lenses, the increased percentage of light due to tne use of the telescope, I = m D£/d2 Bbr all this light to -enter the eye,S,s^,or, substituting the value of d'. from (11) ,d, | D^S. If d,cD/G, as may he the case ifith telescopes designed for special pur- poses, the effective diameter of the object glass Jiill be reduced as far as light is cbncera«d to d, G. This value substituted in the value of of I, gives, 2 2 \ I m B /d , when d, 5 D/G \ ( jgj = mG^ .when d, for 'any two. colors of the spec- . ■ m^^.v^^ tmm bT a oroper relation betireea Jgc^i;.;;^ q-p" ITj the foL? lenlths.readering the oombi.* ->>c4n } natioo nearly achromaiifi.wbile the ^^^^ =-^[JLJv-1 - - V - -• Sharper cnryatore of the convex lens ^..^.V I >-"'* * <• leaves a residual of oouTergmB pie. The finishing of a fine object ^^ss reqaire ^ ^^^^ ,^ ^^^^ kf aSI ^le^nn°?hl''|r?»P M|cted^b^ hh^ S^hVlilesI ^Sa^glTnd^S^lirni^iiVf^lr^Jeieated tests tne -fi«^l^s«lptjl«other^^^^ 'ir.?.Src^r'T5: ^et Lr? r'thl SepLce is usually .ade 'o, us- lLU4lShtiifbftJ°eft^l^=!.ra"sfn|ll%^fw!?f5L-r.uil2!e»^"lo^: oal length, a.s found from Optics. iriere ir..fi',are the focal lengths of the separate lenses, and a is the distance between them. — The Huygenian ,or negative eyepiece, is one of the best irnen cross - hairs are not required. It consists of tso piano convex lenses, Pig. 2S , irith the plane sides tosrards the eye, the. toc^l length of the farther or field glass being 3 times that of the nearer ,^eye-glass. They are placed about half the sam of the fooal lengths apart. The field glass receives the converging rays from the object glass before they have reached the focus, and brings them to a focus be- tween the lenses.. Cross-hairs are often placed at the focus to define _ certain portions of the field.as in M . _ the sextant telescope, but not for ac- H ^P^: curate measurements, since the cross- ?■-«' hairs Bill be distorted, seen through ^ .^^ alxv. c^xV the eyeglass only .while the object "^^ " ^' vrill not be, seen through the correct- ed combiaatioa. Airy replaces the piano convex field glass by a concavoTConvex, increas- ing the flatness of the field. The Bams den, Pig. 25, is the form most commonly used when aeoucate meas - urements with cross balrs or micrometer are required. It IS a positive eyepiece ,i.e., it receives the diverging rays from the object glass after they have passed the focus. The two piano- con- vex lenses have their convex sides turned toviards each other; they have the same focal length, and are placed two=thirds the fooal length apart, giving by (19) an equivalent focus of "3/4 that of one of the lenses The Eellner and the Steinheil are modifications of the Ramsden which are coming into favor on account of the greater flatness of the field or freedom from spherical aberration. In the former, the eyeglass is an aoromatio oousbination and in the lat- ter both are acroinatiojsee Pig. £7. The former has the larger- field. 'Sone of these eyepieces invert the image, and as the abject glass inverts, the objects all appear inverted. Eq.lS) CfiOSS BAIRS. 27 (^cVXxvft* E.t^VY\*«.. Fx<^'xn- SM.VTCkva-^'^ €.^«.-p^iLUK. _1 » \i I-: I' iizii ■i The terrestial eyepieoe oonsists of four lenses, the object being to in- Tert the image so that objects seen through the telescope appear erect auite an appreciable loss of light results fron the tso- extra lenses (at least 14| as estimated by Solan) and a serious shortening of the focal length of the object glass for a given length of telescope which increases the difficulty gf securing a flat field. Tuo combinations are shoin.the Airy aod the Praunhofer. Qiagaaal ^SS^. for convenience in looking at very high objects polished Scecolnir mot.al i^ „1,„.j x..^ ^' 7^ uujouus, yep: fhii azimth. ffor obifiet.» „=;;Vh= „:;::"": rr°.t*°s=/xu axi,i.T,aae mz not in Of the-eyeglgrir^ W^^"?£tt i^e^ifl^? ^ I'VlL'Xc.,, this is HPH ! 1^ ^^^^^-^ ^^^.-i^ tllLl^^J ^' placing the miproc betireen the central lenses of the iri«c"iB'illLih'f ="""'' *''"° ^""^'^^ *"« °''^^°* " »"""<^« a-xi leaves prL°»*ctn °be «3ld%^'?e"ss'nss'of4iSt!^°°''^' "^" '''''' *^^^"^"^^^ 33. CROSS BA1R3. Since ifith the telescope, the image of any point is at the intecsectioD of the focal plane with a line through the point and optical center of the object glass, this optical center may be taken as a fixed point for. all lines of sight. The, intersection of a horizontal and, vertical hair placed in the focal plane (it should be in the optical axis) »ill give a second fixed point. The line joining them, called the liaS af collimation . is taken Tor the direction of the terescope;its greater pee - cision is due to the magnifying poirer and increased light of the instrn- ment. {n pointing, the eyepiece is first focussed upon the cross bairs and then the object ^lass upon, the object: the focal plane of the ob - -jeot glass is thus brou^t to coincide nith that of the cross .hairs, so that the latter irill remain fixed upon the object as the eye is moved from side to side behind the eyepiece. The first is for the eye of the observer, and this focus does not need to be distarbed irhen once properly made;the second is for the distances objeot.rhich requires change srith each ne* distance. Spider lines are * usually nsed for cross hairs. Some prefer to have them spun directly by a spider as needed, others to take them from Cocoons. They should be opaque, cylindrical, free from dust, and so small as compatible uith dis- tinct visibility. Platinnm wires are used by some instrument makers as being more opaque «ind less liable to stretch with age. '.vt. . ,., ,,. -, . The requisite ^fineness is obtainedby coatiag with silver, draning doitn the «ire ancr aftemards removing the siiref by nitric acid. i glass diaphragm irith etched lines is sometimes used in place of cross bairs, nith perhaps some advantage as to perman'ense of position but «(lth the disadvantage of loss of light, and the iia|nification of all dttst on the glass unless thick and the cross hair side inclosed in a sealed iw>>^ The ret icule of wires consists of one_horizontal and on^.^f^Sioal for the ordinary surveying instruments. Sometimes stadia if jres are aaaea For geodetic work the vertical «ire should be replaced by an X\for great- er acearaey in bisecting pole signals. ??LAstronomioal uork./several hoci2ontal and vertical hairs are ased.either eQuidistant or arrangea in .gKwps svmmstrically with reference to the center. The linear dis. tasce 'betiieen the iiires can be computed from the focal length of the ob- ject glass as measured on the outside of the tube to the cross hair dia- phragm, and laid off irith a micrometer. Or better and more accurately, by using a micrometer micros copers an eye-piece and measuring the distance subtended 'by the divisions of a rod at a measured distance; from this dis- tance the required distance between wires is veaaiVi^. compited and laid Off by the miiscajseter jlilowanoe must, of course "be made Jfor the chaag^ SH GE0DE3X. (%34,Fig.28, in focal length foe. paeallel rays. The angular distance can be deter - mined from astronomical observation, or directly from circle readings . 34. TESTS 0? TBLESOOPS. To test for spherical aberratiott . reduce the effective area of the object glass about one-half by a ring of black paper and focus upon a irell defined point. Then remove the ring of pa per and cover the other half of the object glass, the distance the lat- ter must be moved in or out, for distinct vision, uhioh should be small if any, is an index of the spherical aberration. To test for definition .foraig upon small clear print at a. distance. of 30 to 100 feet, depending upon the magnifying poiier,and note if the print is as sharp and well defined as irhen viewed with the naked eye at a distance of 8 to 10 inches. Poor definition may be due to spherical aberration, or to inaccurate curvature, or to variable density or non centering of the lenses. To test for centering , or for the coincidence of the optical axes of the different lenses, fix a white paper disk about one-eighth inch in di - ameter with sharp outline, in the center of a black surface, and look at it when placed in .a good light at a distance of 30 to 40 feet. If the iiTiige of the disk,when a little out of focus is surrounded on all sides by a uniform haze, the .centering is good.: Astronomical objects are sometimes .preferred for testing as follows: the correction ^o.r spherical aberration is well made when the image of a star, undeV'Yavorable conditions appears as a small well detined point or round disk. Having this in the best focus, the slightest motion of the object glass out or in should enlarge the image, it remaining oir - cular if the lens is symmetrical throughoat.jwhile in the most perfect telescopes the image will enlarge' to several concentric rings loicou- lar) of light before disappearing. An imperfect unsymmetrioal lens, will give distorted rings, or only a confused mass of irregularly col - ored light. If the glass is not homogeneous^bright stars will show "Wings" which it is impossible to remove by perfection of figure or ad- justment. The defective portion can be found by covering up differ- ent portions of the object glass and testing. The correction for chromatic aberration is well made, when after focus- sing on a bright object as the moon or Jupiter, pushing in the eyepiece slowly will give a ring of purple and pulling it out, one of pale green, thus showing that the extreme colors of the spectrum, red and violet have been "corrected. The flatness qf the field depends mainly upon the correction for the spherical aberration of the eyepiece . It can be tested by drawing a square some 6 to 8 inches on a side,with heavy black lines upon white paper, and looking at it when flat and at such a distance as to nearly fill the field of view. If the lines appear Perfectly straight tlje field is flat. A telescope. may distort the image appreciably with- out introducing any error in ordinary work, but it is objectionable for stadia work and inadmissable when measurements are to be taken in the field with a micrometer eyepiece. The object glass should be mounted so that its optical axis eoineides with the axis of the telescope tube. The object glass slide shoald be parallel to this same line, and the vertical plane of oollimation should contain it when adjusted perpendicular th the telescope axis. The rear end of the object glass slide is sometimes supported by an adjustable collar for ease in meeting the above requirements, but with first class workmanship it is usually considered unnecessary, while it adds an element of instability . The accuracy of workmanship can be appreciated by remembering that 10 seconds of arc will subtend only X)00049 of an inch forafocal length of 10 inches. The object glass slide is tested by placing the vertical wire in ad - justment for distant objects, (slide drawn in) and then testing the adjust- ment for near ones (object glass slide pushed out). This is of more im- portance for ordinary instruments than for geodetic and astronomical ones where the precaution is taken to not disturb the slide or focus of the object glass between sights, which are combined on the supposition of a fixed line of oollimation. This is possible for sights over 1 1/2 miles long no matter what the ineouality, while it is not for short sights Sq. 19.) GRAOOATED CIRCMS. 29 anless they are nearly equal. { The horizontal line of ooUimation is not restricted as closely as the vertical, so that if it is adjusted parallel to the object glass slide the deTiatioD froB the optical axis of the object glass or iron the axis of the telescope, Hill have no appreciable effect. 35. LEVEL TOSES. These for accurate Jiork are "accurately ground sith emery on a revolving arbor irhich has been turned so as to give the desired curvature. The tube is slouly rotated about its axis so as to distribute the grinding uniformly around the circumference. The surface is then pol- ished ,the tube filled and tested on a level tester for uniform curvature by noting if equal angular changes uill give a uniform motion of the bub- ble, ffor delicate levels, the defects found after this rough grinding must be corrected , requiring repeated trials and much skill and patience . The upper inner surface ,when completed, must be highly polished to render the friction of the babble as small and uniform as possible. The tube should be of oniforir. bore and thickness and of hard glass . The liquid used for filling is usually alcohol for the more common levels, alcohol «ith a little ether added for fluidity for more sensitive ones, and sulphuric ether, ?iith possibly a little chloroform for the most sensitive ones. For delicate levels a chamber is added at one end so that the bubblecan aluays be used at about its normal length for greater convenience and accuracy; a change of length with the temperature changing the zero if the curvature or size at one end differs from that at the other while a short bubble is more sluggish and its position of rest more effected by friction and by local defects of the tube than a long one. The best results Hill be obtained with the length used by the maker in testing the tube. The tube should not be directly held in rigid metallic sup- ports on account of the danger of distortion from pressure due to chang- es of temperature. The support should be at t?(o points only and rith rings of oork or other yielding material which will give sufficient stability. A very sensitive level should be inclosed in a glass box or tube so as to form a closed air space, to diminish local distortion from sudden changes of temperature. th!""* r^J® °* ° division should be determined for different portions of tha the tube to test uniformity.and at different temperatures to toeJm?ne the temperature coefficient if any. 4n appreciable coefficient will asnally denote a cramping of the tube by the supports. ^ 36. aiADOATSD CIRCLES. The process of graduating a circle is essential- ly one of copying the divisions of another circle. The circle to be ooi>- ied is usually some 3 feet or. more in diameter, in which the graduated er- rors have been carefully determined. This is mounted and well centered on a heavy axis firmly sflpported in the graduating Engine. The new cir- cle is placed upon the old, and centered. One method of centering is by allowing the vertical am of a sensitive level to rest against the inner surface of the hollow axis as both circles rotate. The level is radial and pivoted at the upper end of the vertical arm to the fixed frame above so that any eccentricity as the circle rotates sill move the vertical arm .radially and thus change the level. The lines are made by a tool having an automatic cut in a radial direo- tioa.the circle being turned division by division as read by a microscope fiaadabove the large circle or fed agtomatioally by a worm gear acting on the"eireuBf4rence of the circle. Ib the latter case the fear is ad- justed by oarefnl test until equal motions of the worm wheel will rotate the circle through equal angles. This done, the work proceeds automatical- ly with but little hand labor. Bubing this work the temperature must fee kept very constant in order to avoid distortion from unequal expan- sion, With a ten-inoh circle, an ecrcr of 0.0001 of as inch in a division or in centering will give an error of 0.0001 * 5 sin 1" =4.1 seoonds;shiiw- ■Xf6 the extreme acwracy necessary in centering and in graduating a circle 30 GBODSSY. (l§38,Pig. 39, ifhich is to be read to tentns of seconds. . „„„ ,..xp circles and ffi7e-iBinnte spaces are nsaally the finest cat "^P?" ^^^!^,^f °^?^;|°! 10,30 or 30-mnate spaces are the smallest apon ^'■^i^f ="=^^- *°1Se mediate reading are taken >iith verniers o' ^^<^'°fjf^ microscopes, vernier is too well knosn to need a description here. For an illastrated description of the nen dividing engine used b? fauth & Co. of Ifashington.see Zeit .fur. Inst. 1894, p.84. See also O.a.C. 4 G. /■ '3?. MXCROMSTSR MICS0SC0FE3 . These are asnally used in place of srer- ' niers when readings finer than about 5" are required. ,9^°^^^^^"^. f ^f" tached to a frame »hich is n>07ed throagi a bo=i P^^P^"/i^J".'^° ^^!l°" croscope tube by an accurate micrometer screv, vrorkmg against spiral springs, as shown in Pig. 8'9. ri«^.i^. If the mi'croscope has a flat field and the sore» a uniform pitch, the ap- parent motion of the cross ihairs across the limb, will be proportional to the turns of the screir, giving an accurate means of subdividing the spaces on the limb. A common divisim of the limb is into 5 minute spaces, the ob- jective being placed at such a distance that 5 turns of the seres sill move the wires over one space; each turn Hill then give a minute, marked by a tooth on the comb in the edge of the field, as shown, iihile seconds can be read from the head of the screu by dividing it into 60 equal parts. Two parallel hairs are usually used, placed far enough apart so that rhen brought over a division a bright line ;iill show on each side between the hair and scratoh;the equality in width of these light lines being jttdged more accurately than the tiseotion of a scratch by a single h^ir To take a reading,the micrometer screw is turned with the increasing numbers on the head, moving the hairs from zero of the comb back to the first division of the limb to the right(apparent left), the number of teeta passed and the reading of the head giving the minutes and seconds from the division to the zero. Osually the motion of the screw is reversed, turning against the graduation on the head, until the hairs bisect the division to the left of the zero. Only the reading on tne head is noted and this should differ but slightly from the first if the microscope is adjusted' so that 5 complete turns cover an average space. It is often thought desirable to make the bisection with the positive motion upon ohe screw, rather than with the return motion from the spring, to avoid the lost motion. The observer however can work more accurate- ly if free to move the hairs either way to perfect a bisection, than if he can only move them in one direction, turning back and moving up a second time if he passes the scratch. The lost motion will be extremely siiiall if the micrometer is in gooo condition. 4 test of the nearness with flinch a bisection can be dupli- cated by eaofl metaod will decide which should be used in ^^^ given case. The probable error of a single bisection should be about 0' .2. 33. T3E RON OP TBE BCROMBTBR: The micrometer is adjusted ,as stat- ed in ^37, so that the nominal number of turns, usually 5, will move the hairs over a 5-minate space. This can only be approximately realized owing to the imperfections of the micrometer and graduated circle ,the inaccuracies of bisection and reading, and the disturbance due to changes ia temperatarey 2, (20) ■^•i'^^ EBBOHS OP SiADOATEO CIBCLE3 gi ae. correction for ran is nia(te^>e7eral different tays by different ob- servers .jihile many equally good observers regard it as a refinement ihiob it IS a »aste of time to attempt to make. The method given by B.D.Oatts.Asst. U.S.C.4 G. Survey. in App,9 Beport for lS88,appears to be one of the most reasonable. A mean of the first and second readings is taken vihioh averages the errors of bisection and gradaation for the t»o scratches. The differences betjieeh the means of the first readings and those of the second for each neading taken in ob-' serving angles at-t&e station are entered in a Column and added and ^Vs wea* taken for the average run of the micrometer, fhe error in pitch of the screw, due to the lack of adjustment, is distributed proportionally to the lengtb. bet a be the first reading, b.the second reading; r, the average run of themicroueter.tasitive »hen the first readings average greater than thp second. Cbrrection to a = — =c. g SOO' Correction to b - r ( soty .t) 300* The mean ,m = a+b 2 Correction to n r_ (300 -( a+b))4- 300" Correction to m .£. _„ jl. 8 "^ 300 This correction has the same sign as r( = £(a - bj >♦ n) for m < 2' 30*, and the opposite sign for m > ST 30". In the recorS book.the mean of the first micromietep readings is taken, also that of the second. for each reading of the circle, the difference is pit iff thfe r eolUmn and the mean in the m column; after the average r has been fomtd.the correction for each n is taken from the $able II( computed froB (20))snd applied to m iiith its proper sign, giving the corrected read- ings. Por an example, see The Foi'm of Record Book §48. See also the Bun of the Micrometer by George Davidson, in q.S.C.4 G.S. Beport for 1334, App. 8. 89. EBBOBS OF GBADOATED CIBCLBS These may be due to an eccentricity of the upper motion or inner axis viith reference to the center of the gradaation, or they may be due to errors in the division lines themselves The error due to the plane of the circle not being horizontal nhen the axis of the upper motion is vertical as indicated by the levels remain - ing in the center during rotation, is so small in an instrument in shich the limb irill remain flush with the vernier, or the micrometer microscopes in focus during rotation, that it can be neglected. The error due to eccentricity is of more importance with instruments for erdinary surveying irork than irith those for geodetic or astronomical work, for irith the latter all the microscopes or verniers are used in making a reaaing,and it can be readily sho?TO that the mean of any number of eqai - distatct verniers is free from eccentricity. ^ , , . ^. Let 8 be the center of the graduated circle, & ,the center of the axis for the npper motion; EE' the line joining the centers; sT the angle AGE, made up of the index reading z and the micrometeE- readings A,B,C:and e (^-measdEe) , _ the eccentrictty GG*.. For 8 micrometers ISO* apart. e sm z" e sin(180 +z' ) e sin / 1/2(A*B), which is From the 1st. z* = 2 + A f - • 2na. z'. - 2 + B - f = z + B + c Iteaa value , z' ' 7. * l/2(( teee fcoB eccentricity. Por S micrometers 120' apart Prom the 1st., z" - z + A - 2nd., z* - z + E - 3ra., e" z + B ' e sin z'' e sin( la) + a^y s sin (240 +7') (539, Pig.'33, •32. GEODESY ■By 83, sin (133 + 15')+ 3in( 240+/.) = g sia(180 +z'.) 00s 60° =-2 sin z'- > 1/2 ■ =-sin /. 1' ■■ z + V3(A + B + 0), .ifhich is free CiTcV.^-5, -^^ Fxei-3t. .'. Mean 7alae, from ecoeatricity. Similarly it can be shown that the mean of any number of eqnldistant micrometers iflll be free from eooentricity, Sirae instrument makers put in radial abutting ^pstan' head sorei^s between the circle and hollow ixis which supports the upper motion 30 that the eccentricity can be adjusted out before the plate is screwed fast to the flange of the axis. The graduation errors- proper are diyided into accidental and periodie. -The former follow the lar of errors of observation given .in Least Squares, hence their effect is diminished as the square root of the number of lines used. The latter occur at regular intervals according to. some law, and may there- fore be expressed as functions of the reading itself. The sum of all the corrections for periodic error, including those for eccentricity, must;~have the general form * H) = a'. sin('z-fO'.)+u"sia(2z-«J")+u'." sin(3z* a'") + etc. (21) where, "K'z) denotes the correction to the angle z and a'.,0'.,n',O'',etc. , are constants. The shorter the period of any error, the higher is the multiple of z in the term representing it. Ohaavenet, Astronomy .Vol. II, f.52, shows what terms are eliminated by taking the mean .of a number of equidistant microscopes and how to deter- mine Vas constants for a giv- en circle by taking equidis - tant readings around the cir- cumference. R, 3. Woodward, Beport, Chief of EngBS.U.S.A, 1879, Part III.App.M.M., p.l974, takes up the terms not elim- inated by means of a number of equidistairt microscopes and finds their effects upon a measured t of level. Balf the sam of the readings i»ill give the trae vertical angle, and half the difference the index error.- Accuracy of adjistment is of less importance than sith the smaller in—' strnments ased in ordinary sarveying, becanse the observations are ar - ranged to eliaiaate errors of adjastment. Shns if the line of collima - tion is notxto the a3tis,i-t Till describe a cone as the telescope ro - tates; so that in plunging up or doirn throagh a distant signal the li-ne irill not folloir the vertical throagh the signal bat itill cat the plane throagh the vertical perpendicular to the great circle through the points in an hyperbola having its vertex -at the height of the .instrament and its axis horizontal. Ihe horizontal angle aealsuced is then from a point at a distance il, see Pig. ^s.to the left of the section. Upon rever- sal the measorement will be taken from a point x'. to i i . the right. Bat if the collimation error has remain- _Hsi^>B.Ji6^>y^-u.s ed constant and the axis is horizontal,:! will equal , . i X and the error of collimation mil be elimnated by , ^ > taMagi,the mean. ^ , ^ .,.= 'il,' ItTfie telescope axis is not horizontal rhen the v plate levels are in the center, the line th.-ough the ,i distant .signal siill not be vertical but inclined ,re- I ^«^.^3. fecring the horizontal angle to a point at a dis - y\ tanee x tb the left, as in Pig. »« Opon reversal, the ^^^ plate leTOls remaining in the center, the error Tfill ■,•„;„,*„, h» be the aa»« bat in the opposite direction . The mean »:l11 eliminate the error as before. 43. DBTEIRBIH4II08 0? INSTBOIIEIITAL CMHSTAiilTS . Valae of 1* of level . Set BP the iBstrament on a firm support »here it itill be protected from sadden changes ,of temperatare.and place the level on the teiesgope «ith the two tabes parallel. If the tube is chambered, take a tabble of aboat noraa), leagth. Move it' by means of the vertical tangeat soreii from one end of the tabe to the other back and forth.setting at regalar intervals in seconds and reading both ends of the babble. If the circle cannot be read olo«ly enough rod readings at a distance of 108. 1 feet nill give 2' per .001 foot on the rod. gatoe of I?- of micromet er eyepiece . If the sore'^ is nori2ontal(irhioh can be tested by noting if motion of the sores changes the altitude of the horizontal hair) put the micrometer at a given reading and sight to a well defined point by the upper motion and read the circle; tarn the mi - crometer.say 5 turns, and bring the hairs upon the same point by the up - per|motion,then read the circle; continue the process until the desired ac- curacy has been secured. The differeace in the ciraie readings divided by the number of tarns sill give the value of one tuTn for the different parts of the screw. If the screif is vertical, the same method may be employed irith the verti- cal circle if it is suitable. S more accurate method involving more labor is by means of lolloning a ciroumpolar star near upper culmination for the horizontal screw or near elongation for the vertical screjt riith the circle clamped, depend- ing upon the observed time intervals for the angles as described iir Ohauvenet's or Doolittle'a Astronomy in connection with the zenith teles- scope. Wire intervals . These may be determined by the methods given for 1 '^"'•^^•> iieifdOD OP SIMPCiil AOOJUS IffiASUHEMSNTS. 35 of th3 micccmeter. ■Ths circle can be iOTsstigated by the Eethods referred to 539,/(hile the methods for the' telescope have already been gi7en. 43. THS J0THOD OP DISECTION 0KBRMTI0N3 IN HORIZOHTAL ANGLES. This IS the most coiicod method in this country i7ith a direction instrument. k reference line is taken, .vhich may be the signal most easily seen under 73rying atmospheric conditions, or a nark set for the purpose at a suf- ficient distance to aroid changing focus (not less than 1 1/2 miles). The signals are sighted in order around the horizon in the direction of the graduation, beginning' nith the reference line, and the micrometers read for each; the telescope is then reversed, not changing the ends of thH axis in the y's if it ha,s to be taken out for reversal, and the sig- nals are sighted in the reverse order around the horizon, ending tith the mark. This forma a set, and as oany are taken as required. She first signal each time should be approached -jith the telescope from tne same direction as for the others in the half set so that the tendency of the circle to be dragged around by the friction of the upper motion •'?ill be taken up before the first reading. Before each set the circle is shifted so that the readings fon each single object are uni- formly divided over the dhole circle. In order to eliminate periodic error, as pointed out in %39. the circle should be shifted each time ap- proximately 360° '+ mn.shere n is the number of sets, and m the number 01 equidistant microscopes. If the instrument is in good adjustment, it :Till not be necessary to reverse the telescope in the middle of each set provided that the observations are equally divided between the t»o po- sitions. Sometimes the s^eep of the horizon includes the reference line at the end of the first half of the series and at the beginning of the second, especially if many stations are included in the series. This serves to detect instability of the circle. If the instrument has no lo^er motion it is inconvenient to shift, the circle after each set. The 06ast Survey practice in such oases is to choose either 5 or 7 positions, equidistant 350° * 5 or 360° * 7, and take an equal number of sets in each position;suoh that the total shall give the required accuracy. In setting upon the reference line, the zero of the micrometer should be advanced l/n of the smallest division of the limb each time, in order to distribute the micrometer readings uniformly over the space. This -.fill give a uniform division of the readings upon each of the other objects sighted, so that the average of the micrometer readings upon ea^ih object .■(ill be nearly the same, and the correction for error of runs for each angle /rill disappear. The objections to this method of observing angles are thus stated in the N.lf.S.Sur.Report for 1887 by Mr.ililson. "An objection to the method of directions is that it is very difficult, practically impossi- ble indeed, to secure full sets upon ordinary points nheve the highest degree of precision is desirable and where broken sets are decidedly objectionable. In addition to this dranback to the method, another and very serious one arises from the length of time consumed in taking read- ings and bisections to several distant primary stations. 'ben the theodolite, is supported upon a high to'/ier.as is frequently the case, the entire instrument is continually t.iisting in azimuth as the. tcv- er is subjected to the heat of the sun's rays. *[t is therefore of great importance that the intervals between sights should be as short as pos sible and that the tno series in each set should be taken in about the same space of time. Frequently ho.vever, one-half of a set may be taken in five minutes, /ihile the other' may require ten or fifteen". The broken sets are afterwards filled up by new sets including the missing sta- tions and the reference line. 44. THE KETHOD OP SIMPLE ANGLE MEASUREMENT. In this the number of points in each series is reduced to the smallest possible number, or two. The angle betireea each signal and the reference line, or the angles between adjacent signals ,can be measured independently. Or, the measureients namber is takea Flq.lb 36 GEODESY. _ (§45, Pig. 36, can be so arraaged that' tetitsen n, stations o, (ajrl) '■* 2 aagles uill be measared; starting iflth the first station as a reference line and sjiag- iag to the right to each of the others ifill gi7e n, - 1 angles, Pig, 38, then from the second to each of the others to the right (not inclading the first) n, - 2 angles; then from the third;etc.; to the n, - 1 from iihich odIt one an^e is measured. fhe sa-m of the series = first term plos last tero.maltiplied by one-half the namber of terms, = [l«,- 1) + ij.(n,- 1) '* 2 = ■" ( n. - 1) + 2, as stated; above. This gives the same namber oi pointings ,( n, -:l),iipon each signal. Each angle is repeated the same number of times, and this large enough to gire the required acoaraoy To eliminate periodic error, the initial reading for each repetition of an aagle is increased by 360" * SB , as is €43. a being the namber of microscopes and n the namber of repetitions of the angle. To reduce the effect of ac- cidental circle errors, 3chreiber,Zeit. iur Vermess. p.p. 209 -240, 1878, divides the dis- tairee between initial readings for the dif- ferent repetitions of an angle (360 •:■ mn) ly the BBMber of angles, u, - l,to be meas- ured from the first reference station, and increases the initial reading for each ne» angle by this amount, starting from zero. The initihl readings ^or the anglesmeasured from the other stations a.i, initial lines, are taken from the first.using one each time shioh has not already been used with either of the lines forming the angle. An ex- ample of the settings at a station /ihere 6 signals are, sighted may be sce^ in S.Y. S. Report, 1S37, p. 145. This Bethod requires the same number of pointings and readings as the pceoediflg footjo stati<>ns,4/3 as many for 3 stations, 6/4 as many for 4 stations, etc. , provided the visibility of the signals sill alios of al- irays taking full sets by the first method. Por long lines, as in primary triangnlation, these ratios »iJ.l be less owing to imperfect sets by the first method, shile if the delays in waiting for signals to show in order to complete sets' <«« taken into aoooant.the advantages irill often be «ith this method. inOther advantage of this method is that angles can be measared whenever two signals are visible, provided ataospheric conditions are favorable., allowing more time to be utilized while in the field, and each signal to be sighted when untier the most favorable conditions as to illuminatioa and steadiness. 45. THE METHOD OP BBPSTITIOHS. The impression is quite general that this method ulll pot give as good results as those ,jith a dicreetion instrujnent described above, bat. ttie method has been a favorite one with many most excellent observers,'a/id the results obtained have fully jus - tifiad their preference. When the upper motion is always rotated in the same direction, errors due to trist of observing stand^drag of cir- cle by friction of upper motion.travel of clamps, etc., are not eliminated by reversing the telescope, and the resulting angles will usually be too small, although sometimes too large. This is obviated by taking one-half the repetitions upon the angle, and the other half upon its sxplement, always siringlng from left to right with the upper motion. Errors rhi.oh tend to make the an^ too small will thus also tend to make the ex- plemeat too small, of the angle derived from it too large. ^a the li).y,.S. Sorvey the practice was to take tJic;ee repetitions'jof the same angle with telescope direct, reading the circle at beginning and end;then three repetitions of the eixplement with telescope reversed, still swinging the upper motion with the graduation^which is equivalent to "un- winding the circle, i.e., the third repetition will bring the reading baoii nearly to the initial one. The explement thus only enters in the direc- tion of the swing for the'iptKr motion, and not in the figures recorded. They took 6 sets of ft repetitions each for an angle, and the results with 5g.21.) CONDITIOSS PAVCaaBLE FOR 0BSB8V.ISe. 37 only an 8-inoh oirols were as satisfactory on primary iiorn a3~.»nii a di - reotion instrament, ■The angle from a reference line around to each signal can be measured; the reqaired angles then resulting as sums or diiferenoes of the measured ones ifithont the labor of station adjustment;or the angles may be meas- ured as shonn in Fig. 36. The initial readings for the different sets of an angle should differ by SeC/mp as usual, ifhile if the angles are meas- ured as in Pig.Se the additional precaution can' be taken of having no tiro readings alike upon the same signal. 46. C0HDITI0N3 PATORABLS FOR 0BSR7.ING. To support the instrament tri- pod, or stand, three solid posts are set in the ground vertically some tno feet njth tops level, bne for each tripod leg, and well tied together and braced by nailing on boards. The dirt is then tamped around the posts and the center often filled »:itH stone. ;ihen an elevated observing stand is used, sea §33 the tripod or inner tow.er supports the instrument directly irithout the tripod, and the outer tower the observer. In all cases the height of the instrument should be such that the ob - server can look through the telescope when standing erect comfortably Some observers use a more or less portable observatory for the pro tection of the instrument from sun and air currents while observing, but the more comnon practice is to use a tent for primary and secondary work, and an ujnbrella or other simple shelter for tertiary. The tents used on the B.y.S. Survey were octagonal for ground stations and square for elevated observing stands, both 8 feet in diameter, with walls 6 feet high, and made of 8 oz.duck. They are supported by 8 poles, one m the center of each side for the square tent. The wall is ill one piece, sup ported at the top by small pockets which slip over the tops of the poles, with a flap one foot wide at the bottom to tack to the floor to shut out the wind and dust, and a triangular shaped door large enough to admit instrament boxes as well as the observers. The top is in one piece ,held up in the center a foot above the eaves by a rope attached to a small thimble sewed on the outside, with flaps about a foot wide at the eaves which are strapped to the walls, fiiy ropes extend from near the tops of the poles to pegs if on the ground, or to the railings or other parts of the observing stand' if elevated. Floor space is better econ= omized by placing the tent eccentric over the station on account of stor- ing instrument boxes, etc. Care should be taken not to obstruct lines ot sight by tent poles. The walls can be lowered a foot for observing, or a window, one foot wide can be cut around the tent at the height of the eye or telescope and covered by a flap on each side when not in use. Tow.er sheets of 8- oz. duck are sometimes used on two sides of an ele- vated observing tower ,to protect the inner or instrument stand from the wind to prevent vibration, or from the sun to prevent station twist, the exposed stand having a tendency to rotate in azimuth with the sun during a bright sunny day and to return at night. The best time forobserving is on a day when the sky is overcast;next to this is a calm, pleasant, late afternoon; evenings from about an hour after sunset until about midnight are also favorable. The hours for observing upon the O.S. C. S G.Survey are in the summer season, from sunrise until 8 a.m. and from 4 p.m. until sundown. Verti- cal angles are measured from 12 m. to 1 p.m. and in the afternoon until within an hour of sundown- Lines of sight passing close to the surface are most disturbed by heat wave and other atmospheric disturbances, producing the appearance in the telescope often described as "boilingf*. Lines over furnaces and cities are objectionable, while those over bodies of water are not usually so clear as those over land; high lines are least affected by atmospheric disturbances. ifhe readings for an angle should be distributed over different days or divided between forenoon and afternoon, to equalize the effects of ^Lateral refraction, side illumination of signals, etc. No readings aAould be taken under any improper conditions of the atmosphere, as shown 31 eBOOBSX. (^49.ffig.37, chiefly by the appearaooe of the signals. The instranent shoald be hapd- -ed with a light toaiSi and with a certain degree of rapidity, yet in co«= Pieting a pointing it shoald be done oareftilly and deliberately.rithOBt worry or bias as to the resalt,»atohiDg the signal long enoagh to be certain that it is really in the line of collimation aad not temporar - ily there due to oarallaa or a sadden change of refraction either lat- eral or vertical. 47. aCCDRiGI OP eSSOLIS. fhe limiting error adapted by the O.3.C. & G. Survey in closing triangles.ls 3 saoonds for prinary- triangles, 6 foe sec- ondary, and 12 for tertiary. The average errors in closing are of coarse very mtch less. For secondary irbrk,the range of values for an angle is given by Gen. CnCta.a Coast Snrvey authority, at from 5 to 6 seconds, and the probable error as found by comparing the siaparate values irith the mean, not over 9^8 second. These values are given to aid the observer in judging of the accuracy of his results rhile still in the field. On the S.lt.S. Survey the observing party took the precaution to adjust the observations at a station nhile still m the field, in order that ex- tra sets could be taken, or defective ones repeated, in case some of the directions did not sho» sufficient accuracy. The limit for the mean square errar of a direction was placed at V.S for primary iiork,and I'.O for secondary and tertiary. M« FOR.lV\S FOR REtORli, FsTTn. oSf Vltcara. ^s^ Re.T>eo.tiM\^ Ir\»tru.XTv«.tvtS. StC^.'tlotx Tvma I. M«ui ATu:jl.t R«:nu»A4« "fo*- 30'OH'lJ XHI %> 10 3O0M U XS SAT OHlJi OH'«f U It OH HO 0H».1 0H« lO* «■ ISA !«• X6' (jT* S^ocVTjTi .Dg-VS- . OVsgy\ig.-f ■R*= D sin r (22) J^ being so small that sinii =/lsin 1" ( b) Pointing made by bisecting the illuminated cortioo. Bq.85.) BCCENiaiCITX. Bisect the angle FGG,or S.sitbteadiag tbe illaxla- ated portioo as seea by tKe obsecvef. at C, by CE for the line of sight. Then POO ='S'+(i; OCG = (S '(») « sia 1" ' c''* 0. S -(i) sia 1" ' r pio5« a sia CBA C. sin 1" D sin f (^i-) MO CH4PTBR IV.. BiSE ■ LIHES. 51. BASE LINE SITES. Primary bases are from 3 to 11 miles loog.and are placed from SM to 600 miles apart;secooclary from 2 to 3 miles, aad from 50 to 150 miles apart; tertiary from 1/2 to 1 1/2 miles, add from 25 to 40 miles apart. Ttey shoald be so arranged that the sides of all important triangles can be iheoked from a secoad base. If the ooantry is 7ersr flat, the base can be placed anywhere to fit the maia triangalation.bat if rough it ■ay have to first be selected and the triaagalatioo fitted to it. The scheme for coaaectioa oust be sorted ap for each particular case. The small length of base in comparison vrith the distances computed from it, has led >t» T,he attempt to measure accurately, to forms of primary base apparatus. irhiott require a line to be graded longitudinally to slopes of not more than 5° or 6° for a width of 10 or 12 feet , greater elsvat- tloas. being overcome by vertical offsets. 58. EABOy FOBiB OF BASS APPARATUS. Wooden rods vece at first mainly used. A set consisted of 3 or 4 rods,?7hich were placed end to .end begin- ning at the end of the base, the .rear one ras then moved forward and placed in cotrtaot lith the front one, etc. Abandoned at gouns loy Heath, linrg. Ord. Sar,, on account of changes of length due to moisture ,and glass rods sabstltated. Borda Apparatas . Pig. 40. 4 base bars; 2 toises (= 3.898" ) long each of 2 flat strips, upper of copper, lower of platinum, fastened together a-t rear end; difference in expansion measured at front end by graduated scale on copper and vernier on plati- nus at B,' from stiich temper- ature" or change in length inferred. "Contact" by ^^^^_^ slide 0,read by microscope '^^ — " "> 0. Sheltered by board cover above the bar. Strave Apparatus ■ Iron rod wrapped in cloth and raw cottoa. curial thermometer near each end with bulb let into body of bar tacts by contact lever of Fig.41,a spring yielding as the contact end is pushed back by the next bar until the arm reads zero on the scale. Offsets to the ground made irith a transit at right angles and 25 feet distant; the position being held over night by a slide and cube on the top of an iron pin driven 2 feet into the ground. } Bessel Apcaratu s. fig. 42. Components iron and ziao forming a metallic thermometer like Borda's. Sxpansioa,and contact by slim glass wedge between the knife edges at A and B ,the •edge ordinates increasing by 0.01 Paris line, = .0089 inch. Colby _ AcDaratn5 . Ihe components brass and iron ace used to compen- sate for temperature, and not to meas- ure expan?'on as with the Borda and Bessel. The bars are placed side by side and fastened,3t the center as sho.io The microscopic dots,, a ,a' on the «er - Con - - i«n.«i«, TLi*^ 1 >i 1 1 I-; 't l«« 2.2 / ^ _ e s s V y [I ^ _ CoXV^>» a»*e A-p"p».T«v.%^».» TKe^ MV compensating levers remain ■fixed for equal changes of temperature in the two rods. These dots are on toe side Of the case so that the microscopes of Pig.44 can be placed'over them one over Its dot directly, the other over the dot of the other bar by ' The axis in Pig. 44 serves as a tel pushing the bar back for "contact'.' FiA 1\N a e -"-ynwu-i h\ ►-» . V ^•^■> FORRO AFPARAT03. esoope tube for transfers to the groand, its yerticality being indicated by the attached level. The telescope shosn at A serves to align the microscope case. Toe ap-^ per plate connecting the microscopes is brass, the lonar iron, compensating the distance betreen the dots, a, a'. The bar is 10 feet long and the microscopes 6 inches apart. Bz. 3 Find the units of the Borda s«ale,5'ig. 40,sach that an increase of one m differential expansion shall indicate an espansion of l*" per meter for the measuring component. Length for dif. expansion assiimed = 3 .sr B.2. 2. Find the error in the computed length of the Bessel (2 toise) base apparatus due to a difference of 1° in the temperatare of the two components. S3.3. find the length of the compensating levers of E'ig.43,for a dis- tance of 3 inches betreen the two oompoaents. 59. BACHg-ITORDBMAN APPABATJS. (See C.S.B,lB73,App. 12) Length 6"" As seen in the Pig. the ti7o component bars are rigidly at- tached at the rear end to the block A, and supported by rol- lersjirhile the front ends are conaer.ted by a compensating lever B. The con tact rod C projects throagh the end of the case, while the Borda The contact rod E) at the rear end is pivotted at the bottom of the brass. tts inner end knife edge rests against the cylindrical surface G. By Lringiag the '»^^se bar back throagh the case icith a tangent seres, the conr tact rod resting against the rear bar, G,is foroeJ .fringing the bubble x><, fvv«, contact level. H to the center for- contact. »hen in this posi- tion the axis of the cylinder 6 is the axis of the level sector I,so that inclining the bar for slopes does not disturb the contact distances or level so long as the level sector tube remains horisontal. The cross se'ctions of the Borda components are so arranged that, jrhile the tiro have equal absorbing surf aces, their masses are inversely as their stieoific heats, alloTtanoe being made for their different conducting powers., Both surfaces are varnished to give equal absorbing power, and the whole IS protected by a double spar ^'.iwpeAtin case painted white to prevent rapid changes of temperatare. The heads of the supporting metallic tripods are adjusto.'>»Ve. vertically^ laterally, and longitudinally ,the motions for the rear one being control- led by rods running to the contact man at the rear of the bar. Bach tri- pod leg is adjustable by rack and pinion and by fpot screw. The end of a bar right angles 54. POBRO APPABATOS. In this a return is made-to the method of measurement with chain and pins, the base bar taking the place of the c^iain.and 4 mioroT scopes with very firm supports, that of the pins. As originally designed the rod was made of fir, varnished and encased in a copper tube; but as soon ■odified.the fir was replaced by 2 metals, form- ing a Borda thermsDieter. The microscope, Pig. 46, has 2 objectives, one for planbing over a point on the ground, and the other forsighticd at the bar.a cap with a centcal open- cale U can be read through e held in position ninda/( in the by the II levers side. P ,F IS transferred by a transit at 92 GEODESr. (S56,S'ig.47, lag shutting 'oTT the light which does not lass through both rhen I'ooidng at the bar. The telescope of the rear stand is used for alignment by,sighting along the line at an offset target and then aligning the front stand, a scale tak- ing the plape of the front telescope axis. 55. a.S.L.S. RBPSOLD APPASAias. See Pri.Iri.q.S.L.Sanrey, p. 138. This IS of the- Porro type. The components, steel and ^^oo are placed aide by side in a 4-inoh iron tube; they are fastened at the center and are free to erxt pand each nay upon rollers;thsir ends are cat avay to the netitral axes and graduated platinan plates attached. In measuring the microneter micro scope is set upon the zero of the steel bar for contact and a reading tak-r in upon the nearest division of- the zinc for temperatare. The tube stands are placed at the ends of the bar or tube, so that , the front for the first position becomes .rithoat disturbance the rear for the second position, etc. The tiibe is lengthened by a bracket at each end, the rear one resting on a knob in the center of the tube stand bead, the front one carrying 2 rollers, one Wf-shaped,ifhich rest on tracks on the tube stand head. The microscope stand is placed opposite the tube stand, a long bracket supporting the microscope over the end of the bar. The bar is aligned by a telescope on the tube and its inclination meas- ured nith a level sector. ^ j u j ■„ .^ To set a microscope over the starting point,the tube stand tead is re- lieved and a telescope tube placed over the rock crystal toob ?ar king the point Lhe end fitting accurately. The tube is made vertical by an attaoh- Td leiel tube and the microscope set on the zero of a horizontal scale at the top; a direct and reverse reading elimnating any index error of the sMle. The tube is then removed and the end of the base bar brought un- der the microscope. 55. IBANEZ APPARATB3.tEngrg.News, March 1884, p. 133). This is an out - groath of experience in Europe jiith the complicated forms due to the use of the Borda thermometer for ^ temperature or compensation. The bar "is a 4" 110'^ ,ironX-b9.r iiithout case or cover except a large observing tent. Marks are engraved on small platinum disks at points 0.5 apart,; while 4 mercurial thermometers nth bulbs encased in iron filings are attached. (Jnde.rgroundj monuments are set in advance dividing the base line into day-'s jorks.and no transfers to the ground are. al- lowed at other points. Depen; dence is placed u.pon rapid con- tinuous work (160™ per hour, Aarberger BaseJ between these points, and the use of a shel- ter tent for freedom from er- rors' due to instability and to temperature changes th^^f-^'*"^;"^ telescope P is replaced by one having its axis near the object end so that it can be made vertical and set over "\. m. ment at S; next the monu - IS returned, and sighted to a target on the line at A-the rt».»°'if?h°°f n'*^""^ '^ ?^* "P 4-ahead and a target at M,taking ihe place of the telescope axis, is brought into line by sighting through f Its aligning telescope is replaced and sighted to the target aheadj the oar E IS then brought under the microscopes V.,the dot b at the rear end being accurately bisected,w1iile the front microscope is moved longi- tudinally on the slide S to bisect the dot at the front end; the 3rd stand IS set up like the End. and the bar moved forward. Shen a monu- ment is reached a stand is set over it as in starting, the bar pit in po- Eq.SS.) ra.PCiBX BASS APPARATJS. 43 aitiOT antl a 1/2'^ scale used to neasiire the distaoce froiB the niorosoope to a l/S* dot on the tar. 57. U-S.C.S.SS03SD4SI APPARAT05. jBep. lS30,App.l7) The constractioa is olsaply sho»ji in Fig. 48 from SaegBiallar"!s Catalogue. The iiieasariB| rod is steel 4" or 5" long. The oatside tubes ate zino.one fastensd to» steel at the rear lith its Borda scale at the front, the other at the front iijth scale at the rear, iaoh scale is read by a magnifying glass at the top of the case. B is the tangent screif working against the sjrin^ «t the front end for contacts with the slide D. Ttje mercurial themoBater B is attached to the case and its bulb is not in contact uith -the bar. The ease is a pine joist about SC » 8'. The tripods are Bainly of iroodjthe cross bars can be olaitped to the standards at any Jieight. ! S •^T.'ru . -^ ffith the College bars the Borda readings have been abandoned as unsat- isfactory;the case has been covered iilth hair felt and canvas;and the thermoneter has been replaced by tno near the quarter points irith their bulbs in close contact irith the -steel bar and sa^rrounded by iron filings. 5B.U.S.C.&. SilDIROH COMPENSATING APPABATHS.jBep. lS82,App.7) The ej- paasioD of the steel is balanced by that of the zinc for equal temperaHurei changes for the tiro components. The details of the secondary appara- tus, 5 57, are elaborated for the case, coatacts, and tripods Bx. 1. Find the lengths of the oomponsnts of B'ig.49 for a base bar 5" long, fc-j. 2. Sketch the construe tion and find the lengths for a brass and steel oom.bination 6"- iong. 59. S.&.G.S. EHPtBX APPARAiaS. (fiep. 1897,App.llK As seen in Fig.50. there kre 2 separate t;ars irith contact slides, a steel tube and a brass one. They are placed 1 1/8" apart in a brass .tube, irhich can be rotated 180° about its axis in an outside usupportiag tube. In use, double oon- i'"»^«-\ *^t52r" tj- C^ "HS 44 GE0DE3I. (S51,Elg.51, !^ i 1 ii ■* |.i„ _J ^ h .*-w [^ JMw/ e y^. ,*!S'S£*SS^S2l . ». ^ *^ ^ H Ottkflf-ifnf-nt* .. '-] -1- taots are made, steel lo' steel and brass to brass, tbe aocanalated differ- ential eipansloB shoijng itself by the moyeiiient of oae rod apon the other as noted by reading the Ternier and scale at each end at the begin- ning and end of the measarement of the section. About 2 reversals, or rotations of the tnbe.are reqaired per day, ar- Taa'ged syuiiietrioally as to rising and falling temperatare and so as to have the same naaber of bars placed in each position. The oater tube is ooyered ?fith felt and canvas, and the bars are used under a poriable tent drana by a tean as the irorl! prooesds. fc speed of ^Qi S-meter bars an hoar is claiioed to be easily maintain «i. 30. STANDARDS OP tENGTH. ill measiireinents of the- Cbast Survey have been referred to one of the 12 original iron meter bars standardized in 1799 by the French Sommlttee in terms of the toise shioh had served as a standard unit in measaring the meridional arcs of France and Pern. In Sov. 1399 the Sovernment received 3 platinuit iridium bars of the Proto- type meter standardized by the International Bareao at Paris, and from early in 1900 these have referred the Coast Survey standard to the Ihter- aatiorral. The length of the iron bar is noif taken = 1"> + 0.2''± o.ef as the result of recent comparisons, instead of ^ 1" - 0.4'' , as given in 1799. In App. 6 of the Heport for 1893 it is stated that no legal standard of ireight or length was adopted by Congress until July lS68(a Ironghton d2 inch scale hai oeeu used by the Treasury Dept. as a standard in col - leotmg duties, etc.) when the metric system nas legalized and the weights and measures in common use were defined in terms of the metric units giv- ing, lyard fi^ meters; 1 pound j^L- ^g. ^ g^j As a lesult the Survey nou uses 1" 3.^08 1/3 feet .instead of 1" - 3.280889 ft. as formerly. Standards are divided into line measures and end measures : Tilth the former the length is between the end surfaces, with the latter, between lines or points near the ends. 61. COMPAaATORo. In comparing two end measures they are placed between parallel planes br spherical surfaces, first one and then the other, and the change in position of one or both planes measured for the difference in length of the two bars. With the old Saxon pyrometer of the C.3., one plane, iras fixed on the top of the masonry 'pier, while the other, B, was su-pported from a casting at the top of the second pier, being pushed towards the first by a springand held back by a delicate chain C, wound around the vertical cylinder D actuated by a weaker spring. In placing the bar between A and B,the first spring insures contact and the second tension in the chain, giving a fixed position of the cylinder for a FH HS L_ fixed lengthjthis is noted by reading '-^ 8 '-'^ iT? through the telescope E a division of the scale P reflected from a mirror Fie,. si on the cylinder. Slen the other bar is inserted, the cylinder has a different position. and another division is read. The contact level comparator is more coovsnient. especially for fieli. '-'°--'^-^ MSROiaiAL laaRMOMSTSaS. coBFarisoos. A contact le^el is asea at each end to sake certain that ths bar toucties ffithotrt andae pressure. The Tha micrometer seres A is tamed forcing the small rod B against the arir of the level until the bubble reaches the center. The College field comparator has the base bar contact slide in place of the "'' coatoct level. ^^l°Ji°i *^^ ^t"^^ °* " ®"" ^^' ^ ^ "a" ="^6 each compared »itli the standard; they are then placed end to end compared ,ith thf 6-bar in comparing line measures, micrometer microscopes are mounted on piers' ?L S-?/'^^° ^"■^"^ " °'"^°^°^ " distance are freouently reguiredlanf ' the difference in length obtained in terms of the screv? Etor commensurate units the aliquoit parts are marked off on the longer bars aad comparisons made .jith xiie shorter one, the results being added as sith end measures. -^* - e Pig. 5d. sno7s a i"'" comparator used by the International Bureau. The 2 tanks are for determining coefficients of exjansion, one bar being heat- ed by circulating vrarm Hater through the pipes, rhile the other remains ^t a constant temperature. The microscopes shovfU are for readin* the {he'Cffloiseters "feac the bars. The micrometer miorosoopesof the College line - measure comparator can be placed at any distance apart from 4* to 4?" •, ^ j 62r KEBCaaiAL THESiroiiBTEaS. Thermometers are divided into stanrjard and ea-xiliarr. the S'Oales of the Corner include both the boiling and the freez- ing point of nater Thioh allocs of their being studied and standardized - ..-.:.-. --t contain both of by comparisoDT with «ach one indepeirdeiitly: the scales of the latter do not contain both of these'fixed poinrts an'd'they can only be standardized %"ith1lass''!L%"ltf tempered steel, zinc and its alloys, and some other satetahces,th; volume changes lag behind ^"^ temperature changes giving ^i^P to residual eipansjon. This is especially apparent in tbe variation of the JS^tfif^ame of the bulb at the temperature °/ »el^J^| 46 GE0DE3X. (SOT, tig.53, cootraoting is produced much mope rapidly for a given change of temper- atace than the elevation due to sloiijiess in expanding; the rapidity of both moyenents increases as the temperature is raised. Special high melting point glasses Hhe yerre dur of the French, and the Jena of the Germans) are made »hloh haye much less residual expansion than the crys- tal glass commonly used. Uhen a thermometer of verre dur glass is heated from ordinary temper tare to 100° .the stable condition is reached in a feu minutes; *hen cooled more than one-half of the "residual expansion remains after^4 w«. and the stable oonditiott is only reached after several xeete. Ii^h orysr tal.the stable oonditioa at 100° is reached in about an hour,»hil3 months are required after cooling. , .. ^^ . »i. For the accurate determination of temperature, read tne thermoueter.tlien plunge into nelting ice and read; the difference irill give the temperature above 0° referred to the fundamental interval 0° to 100°, tne 100 point having been found by referring to 0° Lj the same way. Small bulbs^are often blown in the tubes of standard lihermometers to alloif of the and 100° points irtthoat too long a t-be. |he scales art the best thermometers are scales of equal parts etched on the steirs. Ihe tube is calibrated by breaking off columns of mercury of different lengths and noting the length in scale divisions asa^•^^'»•» moved from end to end of the tube (a small bulb at the top is necessary for this workl'. The 100* point is coapnted from the observed temperature in steam under- a given barometric pressace.and the 0° point by melting ice immediately after, "this givea.the fundamental interval irhioh is to be divided iato 100 equal pacts for the Cent, scale. The calibration corrections refer these equal volume pacts to the scale divisions^ so that the scale divis- sioDs can be expressed in degrees, h perfect tube and scale nlthin the errors of observation is thus secured and residual expsffisioa can be elim- inated in use. These corrected temperatures (including a correction for pressure on the bulb) ace called meccurial thecmometec temperatures, and they are usually accepted as standard, assuming the expansion oi mercury in glass to be proportional to the temperature. The International Bureau has adopted the hydrogen scale as standard, and by comparing the mercurial thermometer readings with the eorrespoaa.". ■iflg pressuresof a constant volume of hydrogen, by Wariottes la.v, they nave derived correction tables for different kijids of glass. The Coast Survey has also adonted the hydrogen scale. The corrections for verre dur glass are as follons; t Cor. t Oor. t Cov. t • Cor. t Cor. t Oor. -25° 10 ■to. 233 .172 0° 0.000 + 5 -0.0S8 +25°-0.095 +45° 50 -0.106 .103 +65' 70 -0.082 .073 465° 90 -0.038 .036 30 » .102 15 .119 10 .052 35 .108 55 .097 75 .963 95 .013 10 .073 15 -.070 40 .107 60 .090 80 .050 100 .000 5 .034 20 .085 See TIrermomStcie de Precision, by Gliillaume , Paris, 1889. <>5 LENGTH OP APPABATOS. Prom what has been given in4^60-6t the meth- od of finding the length of a base bar is evident. All the comparisons except field comparisons are made in a room so protected that the daily range of temperature is small; thermometers are placed in contact with the bars and a few. readings at a time are taken quickly before the heat of the body causes a local disturbance of the temperature of the bars, the latter being protected by a case or cover. With bars of the same mater- ial the actual temperature need not be knowp very cleaelyi, but the exact difference is essential. Since the probable error in bisecting a line with a micrometer micro scope under favorable conditions is given in S 25 as 4.25*^ upon the retina,* 25 076x26 " O.Wufon the scale ,and 0°.01C changes the. length of a steel bar 0,12>^ per meter, attention should be given to securing good temper- ature conditions, and to avoid the accarailation of constant errors. This will' reouire changing the order of the- readings, the positions of the bars. DEPaCTS AHD DIPPIGOLTIgS. etcfoc th„ different sets. "°'°"" """ i^ir^'iGUbTIgS. 4? The detemination of the coefficient of eixpansion recaices great care on sa^?L~rL^!J'"H°i^*y■°^ ^""^'^fi *11 f«"^ °^ *"« "^^^ ««' at t^e It^rs anf m?rr^L?2f ''?^P"'^ " constant lonl enough to read the therir,oni- eters and nucrometer tiioroscopes. She bar is asuaUy ijjjiersed in ^ater or giycenae, while its companion is snrrounded by melting ice. In Fig. 53 HL='^3h5 ^^ heated by a gas ijet at 3 distance and circnlated through the pipes aho»»; circulation in the tank is se.cured by turning the ,7heeksho»n at the ends. Readings are taken through the water. The comparison of incommensurate units, e.g.,, the foot and meter re quires great care and labor. Comparison of line measures with end measures. field comparisons are yery desirable ,in order to detect any change in length due to disturbance in transportation, ana also to find the actual length of the bar as compared /(ith the computed ,?7hea exposed to sun , ■*ind,and rapid changes of temperature. £x. 1 To find the length of secondary bar No, 1, the folloning comparisons «ith standard Ho. 2 and data, are giyen (C.S.R., 1868J. Dength of standard bar No. 2 at 32° R "" One diyision of the seals of pyrometer Coef. of e;xpaasion for P. scale Thermometer attached to standard ,too high '♦ " » rod Standard Thermo. 77^3 78.0 78.5 77.93 -0.70 So. 8 Piv. 21 15 IS 18 Bod Thermo. 76". 76.4 77.0 76.47 77.23 S799993233 0.0Q000174 0.00000641 "-0:7 0.0 So.l Div. - 10 + 41 t 55 23.67 18.00. 10.67 At 77" At 77 At 77 At 75 77.23 +.76 Computation. 0.76 =< 0.00000641 » 6 = + ^00002923 18.67 x 0.00000174 + 0.00001857 .23 no. 1 longer than stagdapd ,0 .00004780 .23 standard So. 2 6.0017g1R8 . 23 rod no. 1 6.00176968 " ». 1 6.00168391 field comparisons are very desirable ,in order to detect any change in length due to disturbance in transportation, and also to find thl*.;*! length of the bar as compared irlth the computed, when exposed to sun.i"* and rapid changes of temperature. 64. DSfBCTS A8D DlfPICOliTISS. It is very difficult to find the temperatare of a bar and its consequent length under field conditions. lbs Solby apparatus (S52) after being used in England was taken to India ,a l.arge nanbec Ojf bases measured, but the«ompensatioii could not be relied B|»n »nd mercurial thermometers were substituted. The Bessel apparatus (S5£) gaye as the mean of 2 day's observations at the Gflttingen base in Aug. ISSO.the liemperatures shown in 6'ig.54. The case was wood , covered with white cloth and exposed to direct sunlight. Itf If ^■^ ,' 79^^ • -^N _ _• ,-_ .... - — ..^ '^ ^ ^-^ /> <>^ **" "V, / 'A^" f''< ..^r«.^Bru».>'. The tape is paralleled by a German silier . .^ "^♦'^•■•^^ L^ , and by a copper »ire as shOTH. The cur- ' rent from the battery C divides at A, ' part passing through the tape and vire to B and a part through the fine uire AEB. Ti<\.b-s. If the resistance from P to B differs fron, that from B to B current /till floit through the copper »ire operating the circuit breaker D. The arm at E is moved over the dial to 'equalize resistances. As the temperature increases the tape resistance increases (the Herman silver resistance being only slightly affecteo) requiring a/ neif position for E. The dial is graduated under favorable conditions, and the temperature of the tape can then be read under field co"^^'""^- Briensive experiments rere made in connection «ith the measurement of ithe Holton basfc 564) and it ,»as found that the inaccuracy of a tape bJeUne could be reduced to less th» 1/1,000.000, hit at about the Isane cos t as itith bars. ^ In triangulation for bridge spans, or for other work where a fair de- cree of accuracy is required, a 100 ft. tape between tacks in hubs high iffaou.gh to allo.r of swinging freely under a constant spring balance ten- sion will give good results. Hubs 50 or 75 feet apart would give greater accuracy but with more labor. 67. CORasCTIOH PORMOUS. The-length of a base is made up of the fol- lowing terms: (a) the normal length of a bar into the number of times each has been applied: (b) the anoint which the last bar overran or fell short of the end of the base: (c) the amount by which the true length of each bar, corrected for its mean tenoerature during measurement, differs from the normal length, into the number of bars: (d) the sum of the correc- tions duB to contacts (in those forms only in which the distance between.. oonseoutive positions is not the exact length of the bar) : (e) 'the sumoiij the corrections fo^ inclination, both vertically and hori2ontally. "^ Temperatare . The coefficient of expansion is constant between the lim- its usually used, 32° to 100° F, so that the correction can be applied to the mean temperature. Sith zinc, however, a term must be added involving the square of the temperatare. Inclination . Ciet a = the difference in height of the two ends of the bar or tape: b = the inclined length; b' = the reduced length: x = the correction. 50 GEODESY. (SS3,Fig.55, If the iacliaatioQ angle i.is given b' b cos i ; x b-b' b( 1-oos i), op X - 2b sin2 i/2 (27) (27) is best, used by forming a table for each minate ultbin the limits the inclination, using the normal length of the oar. If the average length differs sensibly from the nominal, the total correc- tion for the base can be changed in the ratio, actual mean length to nomin- al length'. for tape lork.ifhere a is -given by leval, b'2 = (b'+ s)8 - 3a = b'^+ £b'' + x2. a2 or X a2/(2b'+ j), a2/2b (nearly; (23) 6B. REDOCTIOa TO SEA LEV-iL: Bass lines are jsually reduced to _3ea lev-i el so that all the computed triangle sides «ill be arcs of the spheroid ;»hose surface is that of the sea produced under the land. [iBt B*" = the reduced horizontal length of the base at an a/erage height h: E the sea level length; y -. the correctioai i^' - radius of curvature of the plane section through the base (see Table V). Then since arcs are to each other as their radii. y ^ (s) Unless h is large, or extreme accuracy is desired the h of the Jenomina - tor may be omitted. 69. ACajRACJt OP RESULTS. This can be inferred; (1) from remeasnrements in segments;(2) by dividing into segmen-ts and connecting the different •ones by triangulation;(3) by (computing the errors from all knoTn sources and addiag. (3) in connection irl'th (l) is the most satisfactory. The principal sources of error for the C.S. secondary bars are: (a) in the length of tne bar as found by the office comparisons, (b) in the ^temperature as inferred from the thermometers, (c) Instability of tri- pods, (d) Backward pressure of contact spring, (e) Inclination horizon- tally and vertically. ( f) -Gontacts.and transfers to the ground. In- the most accurate nork.the probable error frsn all sources is about 1/1,000,000 of the length. It diminishes slightly jiiVa the length. The same expenditure in short bases placed near together will nsnally give triangle sides more accurately than long ones far apart. CHAPTSR V,. TRIGON0«3TRIC AND FSECISE LSVELING. TRieONOMSTaiC LSi^ELING. 70 0BSS3VATI3SS. The zeaitli distaaoes.or vertical angles, are usually fflsasareo «P.sn the station is oooapied for horizoatal angles. Sais Bay ba done »ic!i a vsctieal circle, or differences nay be ot>tained ■,coefficients of refraction: 8iC_,angle of refraction. Assuming the an gle bet?reen the tangent Ij A and the chord AB.Fig. 56, proportional to the distance is equivalent to assuming the line of sight an arc of a oircle°),V though the actual curvature is irregmlar. (a). Non simultaneous observations : BJ^ormula 26) hp-^h, = (h„ + hi+ 2a ) tan ffA-B)/^ ^ 2 1 Z ^^^J-f.^^g)/;! Bui;, A= 130° -J,- ffiC ; B 130°^5»- mg C , giving (A-B)/2 (5^-5, )/3 +C(w:^-mi)/2 Prom the A AEG, (A+B)/2 =90'- C/2 Substituting, hg - hj = tanC(^,.-S'J/23+ CfiDg ' mj )/3 (hg+h^+2ll^)^o»vC/^ C/2 being small, tan 0/2 = k/ER^ + k^/24 R^.ty Formula 15l- CCnn^- m|)/2 being very ssall, Formula S) Si'es, , . tan((S'^-^. )/2 + C(m^ - n,)/2)= tan (.K'^t '/2 + k(m^ - m,)/2 B, Stttetitutingj^^^ ^ tan(''*«^l«> (■b). Simultaneous observations :m, - m^.giving. "h^ -h, = k tariliv-a-, )/2)(V+ (h^ + 5;T/2 R^'* k''/l£Ri irnioh is the formala used on the ff.S.C.S S.S. fsor 52 GEODESY. (§74,Pig.53, (c). tenitli dis tapes at one station only . k-lSO^-S,- B.Cas before . B =S,+in, C-C;gi7ing ik-EifZ - 30°'{S, 4ij..5)C). h^^hl^'^rZT (5;+fin^..5)K/Sj3in 1") (l+fttg+hi )/a^ ♦ iViaP^) .(=0 ' By calling the second factor unity and expanding the cot by Poriwla ^J.M can be reduced to another form «hioh is sometimes given. tan(?.4U,-. .5U ) H™.-5)C l^^^{l^^ S, H™, -.^)'' " ootJ,+(.5-m)C *{.5 -a) oot^S'.C.by Formila 32) Substituting in (31), hg . h^- k ootS, +f.5-m,)K7a^ *A\5?'^i)Jl^:lH<'^*^^ <22) It the line is sighted from the other end.a second yalue »ill be ob - tained.and the weighted mean will give the required result. 72 OOBPPICIEHT OP RBFRACTI08. " Prom Pig. 5B, 5,1 iii,C +Sia^O = ISO" + CjOT m^ + iii2=(l80' ^ (S,+ S-0)CR, /k )sin 1' +1 (33) The refraction coefficients are thus indeterminate from any naaoBr of reciprocal observations, since tsp unknOFnis are introduced for eao6 equa- tion. If the observations are aimaltaneoos, b, is usually assumed equal to m^ . each line sill give a value for m, and the average for the »hole area can thus be fbund by taking the ireighted mean. Thus m = (130° - {S.^S^ )) (R^/81c)3in 1' + 1/2 (34) If not simultaneous, the coefficient for the lines radiating from each .station may be taken the same, so that in a system of 1 lines joining p .points, there npuld be p unknoBD coefficients with 1 observation equa' tions of the form (-33) . If l>p,the coefficients houIc be found by a least squares adjustment. If the veighl of each 5 be taken proportional tto. the number of observations,!!, then by Part 1, S3 and (.W-ji, (33) #onld have a weight w given by 1/ir = Risin'- l'/(^ , ('Vaj+l/n2)/k' (35) that is the nei^t would be proportional to n^ Ug k'^/(nj^ +n_) Bessel assigns weights by the arbitary formula, n^ lei's" /(d. + Ug) OB the ground that errors arising^ from variations in n ar e of more im- Ijortan'ce than those from errors la s , The average value of m as found by the tJ.S»C J G.Survey is: Across parts of the sea. near the coast, 0.073 Between primary stations 0.071 In the interior of the country, about 0.065 Clarke, Geodesg. p. SI, gives the range in India from -0.09 to +1.21. 73. OBSERVED ASSLE OP E[• + C, or C ((5, r 90°)/^ - «i. ) 5,+ 10,0 sabstituting, (a, / 2)/(l-B,)'^ tanS iS,. 90'} (38) 76.IHSTaBllBHTS. Precise spirit, or geodetic, leveling is distingnished from ordinary spirit leveling by the ase of better instraments sati meth- ods and more care in observing. Some of the more common instruments in ase are shoirn. in Figs. 58,59,60, the level is ased as a striding level giving great- er facility of adjustment for both level 'latbe and oollimation,and oppor- tanity to eliminate both errors by reversals in observing. The rear y can be raised or loirered by a micrometer seres, giving a delicate means pt releveling uhen pointing at the rod. In Sig.eO, this slight relev- eling cannot offset the H.J. of the instrument als uith the others. In Pig. 61, the level -tabe is dropped into the telescope tube doirn to the cone of sight rays, in order to diminish the lack of parallelism of the 2 tubes due to locally heating either end of the instrument, thus sacrificing the striding level. The tiro tubes are cast from an iron - nickel alloy having 9 coefficient of expaasion = C. 000, 004 (Cent.)', a- baut 1/5 that of brass. The motion nith micrometer scresr is retained . In Figs. 59 and 61,the mirror for reflebtiag the bubble to the obsec ver at the eye end is reolaoed by a system of prisBS iitioh eliminates parallax by giving vertical sigit cays npolr both ends or the Babble. Fig. 59 has a qriek leveling ball and socket tripod head which is very stable. The fooossing eide of the telescope should be long and rell fitted to preserve parslielism with the line of oollimation irhen sighting at different distances. Bnff i Berger have a more recent type of Fig. 60 la which the level tate is placed on top as a striding level irith a mirror above as in Kg. 58 rather than at the side. The pox^r is 50, iritb 2T1 level di- visions. The principal instramen-tal constants are Fig. Focal length Biam. of -Objective. Power Stadia ratio. Two m.m. div. of level. 60 81 14"l/2 14 15 16 I 1/2 in 1.4 TVs 1.7 . 50 25 35 i3v32 1/231 V 100^1/; 1/100 1/333 1.7 to ^U !00 8.3 6.1 to 8 < a It will be noted that these values do not differ materially from those tor ordinary levels, except in the sensitiveness of the level tube and inr the magnifying power. 77. BODS. Botk target and speaking~aoD -extensible rods are ased.. 54 GE-OCiSY. The Kero or S»»ss rod is slio'n id !^i?.62 Eaflrs. Corps uttti the Kero leoel. Toe smal meters, «bile readings are esliuated to mil Tli3 frencb rod is sboirtr In Pig. S3. It tias printed opon paper and pasted to the rod. To determine changes in length due to oban ture an iron and a brass bar are inserted s line and fasten^e^c^ to the base plate, while ed to the bras3°'2!ne to the wood, each being The brass scale is so. gradjated that each (S77.Fig.65, Thia- IS ased by the U.S. lest'iradaations are centi - i meters. a line graaaation to 2'"'" The rod is rather flexible, ges in temperature and mois- ide by side near the center at the top a scale is-attaoh- read bjr an index on the iron, division represents an expaa- n— ^ I i r^ sioo of JO"''"'per meter of the iron bar: • each division of the scale on the wood gives an expansion of 10'' per meter of the wood. The sam of the t»o readings. (a and E) sill thus- give the total change in length of thd ifooden rod. The a.S..G.S. rod shown in Fig. 64 is a doable target rod made by ». 4 L. E.Gurley of "hite pine impregnated ?rith boiling paraffine to a ! depth Qf,'l/8". It is graduated on both sides and each has 2 targets, one oval and red, the other rectangular and black. The targets are handled by endless tapes as shown, the Tengtb of the rod being a little over 10 ft. The. steel base shoe has an, area 1/2. o\ a. ^t(^^J.■.l^..n«.■^^. The no targets are for use on "double rodded lines," nhere two sets of turning points and tiro sets of notes are carried through sith one in - strumeni.the insinimeat lio set.ting the rear and front rod targets as usual for the first set of T.F's, then the' front ana rear rod targets of iihe T?the'r faces for the second set; afterwards checking both tari^et readings as the ins'trument and rear rod are moved fomard. The U.S.G.S. speaking rod is shown at Pig. 64 a. The unit is O.Z ft, divided and read to fifths. or to, .004 foot. The notes are kept oa tvt g-ft. basis to correspond .requiring all derived elevations to be d'oub- '^^•''°-' PRSNCe G3».. METHOD. 55 bled The shaded portion is red. the other pootioa's black. on a white grouna. The U.S.C.4 G.S.rod is shorn in ^ig.as. The centimeter gradnations lii-^ on the edge 2. 2'"" uide. The center of the tell netal foot is in the plane of the graduation. Silver faced plugs are placed 1'^ apart and the distances between them '^S*?^! "^i "^ ^'^f^ tape for field comparisons. A thermonieter is attach- ed tor temperature. and a disk level for plumbing as irith the others. The pine is soaked in boiling paraffine for its entire thickness .viiich increases the neight.does aiay with moisture changes and does not ap- preciably affect the ooefficientoV t-spixTviio-n th!®;o»;^;.,S:®?S§' ^'^^^S U the instrument is leveled and pointed at the rear rod; both ends of the bubble are read. the 3 «ires and the lev- el again. for the backsight. Similarly for the front sight. The length of sight is limited to lOO""" . and the difference between front and tack sight to 10"" . A heavy canvas umbrella is used to protect from the sun, or sometimes a tent if t*ie neather is windy. Each rod reading is corrected for observed inclination by the formula , Correction 4 id(lm tan l"/4) 4 i d a (39) wiiere 4 i E + E', r (0 + d),the sum of the t*o eye end minus the sum of the tiro object end readings of the levelj d = stadia interval; X - value of 1^ of level; m = stadia constant; A Im tan l''/4,a constant. The difference in elevation between two B.Ms, is corrected by the tor- naia. B. ki. Cor. « {.d, Td^) cm tan 1" (40) where d, suii of stadia intervals for back sights; d^ - sum of stadia intervals for front sights; o = inclination of line of collimation in seooads when the bubble is in the center (+ if object end low), c must include ioeguality. of pivots, level error and collimation error. The level tabe is adjusted until within 2 divisions, and the oollima - tioo until the mean of the The level tube is adjusted natil within 2 divisions, and the collima - lion u-ntil the itean of the 3 wires for direct and reverse position up on a rod at a distance of 50'^ do not differ mors than 2.5""". Read ings are then taken every morning.and at other limes when there is rea- son to suspect disturbances, for the level tube and collimation errors to use in (40). By keeping the saiiB of the stadia intervals, as in the record shown, these can be made equal in closing on a B. V. so that ihe correction (40) will disappear. Steel pins are fceqaently used for turning points instead of the foot plate of Fife. 62. FORM OF RECORD Da^q. Dvr«.eLt.\ovv BACK. SIGrUT 0\,S«.-rM-1^ n'Vi". IUlIV sn tost ■^ aM03 - 1131 0S%!>' 34!.' 10 • RsXt !<.<. VtVf - -XHOiT - a *\ + 1191 l»*<. 34S T P 3 311 0314' -t'0<^«^ - 1441 lyoM 31.-7 4" A'''*T,G> = 3!<- ■31» lOM -0it3 + 3 -1 + l'8!n T-ais 34-7 1. nft-3t A-*B= >i-'>- i-u-n CtO^THS., n 1 3 yo To to\» 4ft 4 ST i «.o-rr«.t. Hi- 1M4 D-¥«.->o ^D = --in lib- D-v«.<0 -liT^Sii <».= -300 Re^ -TC^Th^i'tw^'ft be oeL<.©T-r«.«-tvO'rvS to i**"-**^- 80. tr.S.S.S'. METHOD, tor double rodded lines and the double target rods of Fig.64,the rear rodnan holds on the T.P. of line A and olaaips his red target nhen covered by the cross hair; the front rodmao then holds OB the Dsst T.P. of line A and elamps his red target at the props ec height;he then holds on the T.P. of line B and clasps his black tar- get, the rear rodnan then holds on the rear l.P. of line'^aad and claaps^ his black target. ° Separate noteS' are kept for the its, lines (claimed to se sQiivalen- ta tiaViag been ran in opposite d^Tsctions): irhile the IhslrLineat man ' checls all 4 rod readings as he and the rear rodnan sove forward. The babble is kept in the center irhen sighting Steel pegs are pre - ferred for LP's. The level is adjusted daily, or oftener irhen aeoessary. Attention is called to the fact that the length of sight should ba kept so nearly constant that the focus of the telescope l A 1 1 ,i- . ^■ygtt^ •i.c fty^ SMudoty Mo^x- Ta.T. Rod. i«v«X Tow. tevr^tik X -0.1. CSbO D R O or. n.Q^B To take a reading; a^ Ihe babble is- broirght near the center and the target clamped to correspond,the babble is then accurately centered^and the micrometer scretr of the rear 7 read; thp. target is •bisected by tarn- Bq.41.) IHEfflALITY OF PIVOTS. 57 ing the miorometer sorer aird the screw again read; b. The level is re- T.ers-ed, the hibble brought to the center, and the target biseoted.and both sore»' i-eadin^ taken, o. The telescope is rotated 180° about the optical axis, the bubble brought to the center, and the target bisetjted, and both screw readings recorded, d. The level is made direct, the bub- ble brought to the center, and the target biseoted.and both sore/r- read- ings recorded. The stadia hairs and the edges of the target are then read \>j the le\r- elman; while the target and the rod thermometer are read by the rodman. Having the valae of 1^ of the micrometer screir.and the distance to the rod, the rod correction for each of the 4 readings can be computed by a formula similar to (39); the average of the 4 added to the target read- iBg will give the corrected rod reading. The method of double rodding.is in use, as also that. of running a sin- gle line through and checking back. In the neir method introduced in 1899. and Slightly modified in 1900 to adapt it to the neir level, Fig.Sl.the babble is kept in the center irhile' reading the 3 irires to millimeters on the speaking rod;the front and back sight readings are so taken that the time interval between shall be s4Ball;at odd stations the back sight is taken first, and at even sta- tions the front sight; the difference betireea front and back sight dis- tairoes- iS' limited to 10"; the diffenence between sans of front and back sight distances betirpen any- 2 B.H'Swto 20'";greatestlength of sight 150". Tbe cheek line is oanally^ roil in the opposite direction from the di- rect, and aater different atmospheric conditions, e.g., one in the forenoon the other iir the afternoov] a differeirce > 4"'v€iitanoe~iff~Hloneters. betireeir adjacent B.ITs.calls foe the rerunning of both lines ontilS val- ues are obtained withiB the limit. The rodman reads the rod thermometer each time, and a temperature cor- rection iS' applied^ The error of colllBatiotr is detaraiBed each day b? ssing a front sight neading(after completing a set up/ with a ner back sight reading about ID*" behind the levelrthen setting up about 10" behind the front rod and reading both rods again. The correction constant. ' correction/Cstadia interval), is found by ^^,,■J (suB of near rod readings)' * (aan of distant rod readings) \ (son of distant stadia intervals) -( sum of near stadia intervals) no adjustment is made unless C^ 0.005. Correction is aade for curvatnre and refraction .and for level when the stadia intervals differ for front and back sights;also for length of cod. FC !»R^A O^ H e.co^Ti MSAn TVnoA rota. ^A««l^ fell m3 oaas 1 03\ 1 lis 1 oao.'i 93 M0« 3S x!fi«je too 1S9 I03 I03 \ot MOS The.correetions bets;een ^fi^Vs. are sanmed fhom tables or slide rale iSDd entered on the computation sheet separately. 82. IHBOIALIT: 09 K70TS. The level is set ap ta a pier or other firm support t^ere it is protected from air earrents and from sadden changes of temperature and the babble brought to the center. The telescope is changed end for end in the 7's.and the babble read without reversal. The oqt of level, if any.mnst be tiricedrithin the errors of observatiOB) the ineqaslxty of pivots referred to the supporting Y''3.,ot 4 times the error referred to the telescope azls on the basis of circular collars. ^8 GSODESY The obser^atioas should be repeated cared. uatil the (^83.?ig.66, desired accuracy Is se- Fck.-vi.vV P-v a.tl%«. V- eve"\ , tVXo^v.Hl.l'i^o NNox^ .w.V.-v.^ov, u«.wO. £o.S^^ a.TVci. e.>\«. «.Tva. DOfR Cxv%^ \M«.it ■n.-iA'K '^-^'- VJ D M1.H «.»- -'■"" -.XT oS, R. <). HH -l.s-0 E D n.-v HV .1 j-y R. H3.\ «.l. .1 t i% vo D HO.* »■,» -'■> -l.Ml R I-Si- HH.H -i.'^s- .■74- E o l.t -lis > -.-M R. lis 1.9 -.34- .SO w D HO. if Jf.s- -'■■ - 1.11 R 1.3 HYH - l.«i- I.CK. E D •7.C HV-' -x.tx R. q.% Its - xii- ^ 11% E D I.T ir.T ■^ 1»- R H3.0 «.o .ir 1.00 w o ■».■» s.» yn .,,,- a «!.l." HH.W E D •l.b- 11* 1,03 , i^ H%.q 11 .1 t>A««Mv'= 1,01 McO-v^ft. ia-o;^\-a.v)«X=.v'Si. Referred to. telescope axis. Bye end large l.Ol ' 3.8/2 = l'C92. Ctop- rection to rod reading oegative. Bi.l Jf the collars are IC apart and the angle nade bj the sides of the Y supports acl le»el legs are 90°, find the ineoaality of the col lara in inches for the valoe l"92 giiren above. 83. BOD CORRECTION. For the paraffined rods and those ffliere a brass »oale is ased.the temperatare at which the rod is standard can be found by oOBparison irith a standard. A table of double entry can then be made out or a slide rule used for the correction for any observed temperatare and rod reading, it being the product of the temperature inorement.tbe ro4 reading, and the coefficient of ess pans ion, and positiye »hen the rod is too long or the reading too small. For the Kerp rod which changes length with ir.oisture as well as with temperature tne actual error per unit, length can be determined from day to day by comparison with a standard tape and tt^p corresponding correc- tion applied if appreciable. Par the French rod the paper scales rors.and the iraooeii rods corrections for length, and for changes in length as denoted by the A and B readings. ThiS' is accomplished by comparing the rods irith a standard and at the same time reading the scales & and B. The scale corrections are platted on cross seqtion paper as ordinates irith rod readings for abscissas and the correction curve drawn. An equalia- iag line is- also draun through the origin, which separates the correotioa into 2 partS'.one proportional to the rod reading and the other a local scale correction. It is assumed that only the first is affected by a • change in tite length of the rod To obtain the corrections graphical- ly. the straight line correction, say 150*"-^er 1-" for A + B = 135, is laid off on a vertical from D.Pig.eS.to a scale 15/1, An oblique line Is drann through D and these corrections projected upon it by horizontals, and the oorrss ponding rod corrections macited. If the rod should expand or contract l'*'"'"per meter, the inclination reqiUce jxi.crectiOB tor scale er- Ai fiq. 41. ) ACCURACY OF RSSOLW. 5-, Of DP can be changed so that the projected length coccesponding to the 1st. meter shall be 1=^'"'" longer or shorter than before, ?fhen the cor- rections' will all project into their nei values. The cosines' of the irew inclinations of DP for Twines changing by 1* will thus differ by anity for the radias 150. .-.describe an arc sitb D as a center. lay off the different angles found from the cosines starting from the vertical and mark the corresponding numbers for A + B. starting ulth the highest expected in the field irork Then irith the scale correc- tions as radii and the corresponding points on DP as centers describe arcs. Horizontal tangents to these k.aros »ill give constant values to these projected scale errors. »hile the straight line correction ?till de- pend upon the A and B setting. The corrections for the other rod are placed on the same sheet irith the center at S. A celluloid sheet is ruled with S*™" lines. to the scale 15/1. and kept in position by the strip 91. To take oa.t a correction for a set up; set each arm to the correct A .■4- B; slide the celluloid until the zero coincides ulth rod II read- ing and read the scale for rod 1 reading. Thus if ('A -e B)i = 135; (A 3+' B)i = 118; the correction for a back sight reading of 2.0 on I and a front sight reading of 1.5 on rod II irould = + 108^"'". 84. ACOORACT AND COST OP BESHLTS". The authors of dever des. Plans et Nivellement estimate the probable error for a set up iiith the French Gov. level for sights 75" long as folloirs: 1. Brror of level. The eye can detect a difference of 1/S""ln the readings of the ends of the bubble with the S^'divisions on the tabe. ThiS' gives a probable inequality of about g^^^or a probable out of level of 1*""". This irould give the same uncertainty for a rod reading at a distance equal the radius of curvature. or SO"", or 1.5*""^t 75'^. 8. Error of estimation. iJith a power of S.the centimeters of the rod at 75" appear of the same size as millimeters at 0.3^ Under .^^ these conditions tenths- can be easily estimated with a probable errSr^'^ 0.33^-"'", giving 3.3*"""when referred back to the rod. 3. Errors due to temperature changes. Bxperienoe has shown these to be aS' great as No. 2. Sombining, the total for a reading. r 'v/d.S)*-* {Q.sT* (3.3f = 5^'"'" For a set up.^. P. to T.P., r'. = Vr->- + r'^ = r s/S With 75'"sights. there are 8 2/3 set ups- per l'^ .while with the 4 ob- eerved differences betneen each pair of T P's would give the resulting probable error per i«- , r^ = r VZ X g.66/4 = S^"^"^ wtioh agrees- with the results found for the fundamental French lines. The above supposes- all constant or systematic errors eliminated by the methods^ of observation or by applying computed corrections. The principal constant errors recognized are: 1. The variation of gravity with latitude, fhis results in makinc) the distanoe between 8 level surfaces vary inversely with g, the work re- quired to raise a unit mass from one to the other. or hg, being constant. The observed difference in height of 8 points would thus depend on the height of the line of levels ran between them. Heights fabove sea lev- el obtained by direct measurement are called orthometric, obtained on the basis- of work done in raising a unit mass. dynamic; the differences are usually within the errors of observation, but in rugged, country they may be greater. Por full discasedon see Helmert Hbhere Geodesie. or Lever des Plans . .. . 2. Variations of refraction with height of line of sight, with charac- ter of ground surface over which the line passes, and with the time -of day. In as-cendiag or descending long grades this becomes cuiiiij.ative and may easily exceed the accideotal errors unless short sights are taken. 60 GEODEBJ. (S87,Pig.66. 3. Caangs in height of instrimient or T.P. due to satUement or spring- ing up of ground. This has long been one of the reasons assigned foe greater discrepancies betw.een lines ran in" opposite directions as com- ^red with those run' in the same direction. 4. Change in oollimation and le^el error due to heating the end of the telescope nearest the son-, fhis is the principal reason assign- ed by the Coast Surrey for the change in method introduced in 1S99. In Proo.4m.Soc. C.Bngrs,7ol.26.p 888, the prob. error per kilometer is giyen- for some 1200 miles of tJ.S'.O.i G. levels averaging 1.07"','and for some 1500 miles- of U.S. Engr. Corps levels averaging 0.63r" These ap- parently are from circuit closures. In checkia'g forward and back between benches the limit = 4" N/kilometeps. as already stated. The cost is. estimated by D.ltolitor (iPro.i.S.C.E, 86, p. 897) at $24. per mile for a double line with permanent bench marks about 0.6 mile apart. On p. 1160 it is stated by Hayford that the total cost of the 1899 work of the C.S. was'lS.SS per mile. Seven minutes per station is given as about the average time, for the same (C.S.) work with a record of 111 stations in 9^- 20"" on June SO, jiith 40™ to 80'" sights. and of 10.3 miles- July 14 in 7.4 hours with SO'" to llC" sights. 85. DAHIM. Mean sea level is the ultimate datum to which all land, levels should be referred. It can- be obtained approximately from the mean of two consecutive high tides-. and the intermediate low tide. For more accurate results, a permanent bench mark and a tide gage should be established and readings taken for a semi -lunation .or longer. The zero of the tide gage shou-ld be occasionally referred to the B.M. to guard against disturbance. The yearly meanS' of six year's observations at Sandy Hook, with a self recording gage, gave a mean which has a probable error of 0.031 feet; the lowest mean 1876, being 0.168 below, and the highest, 1378,0. 177 feet above. ^^ CSAPTSSi;. I. TOPOGSAPaiC AHD 3YDR0(BAPaiC SURVEJIIIG. 86. TOPO®iAPHIC-SURVEnHG. The problem is usually to coUeot the greatest possible amount of reliable information for a given expend! - ture wbich shall at the same time bring out the characteristics of the entire area irith a detail proportioned to their relative importanoS and the objects- in view. Tnile the methods are mainly those of ordinary surveying. the young topographer soon learns to distinguish the difference in accuracy and detail required for an exploration survav and a survey of valuable property for the proper study of proposed improvements. In exploration- surveys, check points are obtained by observations for latitude and lon- gitade; in more detailed surveys covering considerable areas the best resalts. are obtained by starting from triangulation points, only a few miles apart, whose positions are known both horizontally and vertically. Method with transit and stadia; plane table and stadia) preparation of plane table sheet; n-point problem: Colvin's lake meander; baromet- ric heights; aneroid profile; Ashbuftier's method with aneroid; photo- graphic methods ; sketching. Only such details should be taken as will show wben plotted to scale. Small distances which can be estimated as closely &s they can be plotted need not be measured. On the other hand, mistakes-.omissions, inaccuracies, etc. .which arenot noticed by the inex- ' perienoed who have been over the ground, show themselves when the map is put to use. or are often picked .out, and the map condemned by soms' old resident who is familiar with the particalar locality. 87. HTOROGRAPHIO SU8VEYIHS. River Survevj!; . For the best results a triangulation should first be extended along the river valley, and con- venient points established for the detailed survey. Otherwise, points can be fixed by latitade and longitu.-de observations, ffor a small stream a traverse line can- be run along shore, the width can be found by di- rect measurement, by stadia. or by bearings from two stations on one s"Kot? Eq 41 ) MBRIDIAS SBCSIOH . ^* to k pilot OQ the opposite sdore. If tHe baffks are Impassible the mean- der line can be ro-Ton the water.iisiag a boat.the distances being obtain- ed with a 'long chain or irire, or by stadia. „,„„„, Depths, oross-sectioBS. character of the hottom.Telooity of the eurreat . volume of irater.rate of its surface slope, and high and loir nater marls are often important. ^ --.u , „»«._„ _m»i, Per a navigable stream.the traverse line may be ran »ith d steamer »hioh may be steered by a compass or by 2 points in line ahead. The direction should be changed oaickly so that the course »ill be made up of a series of straight lines. ' Distances along the line may be measured with the log.anchored log.or buoy and nipper. Bearings should .be taken to side objects by an observer on deck from tno or more positions, and the time of each noted, fbe sketch must, of course, be kept up as the vessel, pro- ceeds. If some distant prominent object can be sighted frequently it villi serve as a check, on the bearings.. Scundiags may be takerl irith a common lead, unless specimens of the bottom are required. T«o boats can be used in place of the steamer. The distance betneen them may be found by the angle subtended at one by a mast of known height at the other. If triangulation points haye been established, the boat's position can be tied to them as often aa desired by the N- point problen.oc by taking, cuts to it at a given signal ,irLth transits at 2 or more stations. If approjiimate latitudes and longitudes are the only checks, only rough nork can be expected. In all field work the day's notes should be carefully looked over at Amht.and plotted if the work is to be plotted, so that all mistakes and obscure parts can receive attention vihile the notes are fresh and the parties still in the field: also the better to lay out the remaining sork nith reference to that already done. Lake . harbor . sea coast, surveys. General methods; methods of locating found- ings. A lide gage should be established and records kept so that all shallow soiindings can- be reduced to loo water. The position of the chan- nel;charact3r of the botton;depth3;and for approaches to harboTs,vie»3 of the shore as seen from different points with "ranges" and angles be- tween prominent objects; are usually required. Lead with tallow for specimens of the bottom. Sand's specimRo cup. Erook's specimen cup. Ericsson's lead, American method, 32- pound shot , not recovered. A wire is used for the line in very deep soundings, and the instant of striking bottom is determined by the change in rate of de- scent. MiUer-Oassella thermometers for deep water temperature. 88. PIELU CO Mill N I CATIONS. Ifith several parties in the field, it is some- times very convenient to be able to cummunicate ^th each other. Vae (brse telegraphic alphabet is usually employed. For long distances •iie heliotrope is used for flashes, the parties having order* to watch for Isignals at a certain hoar each day. For short distances a flag is used. C H 4 P T B H 7. I I. PIGDBB OP IBS EAfiTB. 89. MERIDIAN SECnON.COOaDINATES OP POINT. In reducing geodetic da- ta the earth is usually assumed to be an ellipse of revolution. The dimensions given in Table I best satisfy the degree measurements wtich' bad been mads up to the time wtieo they were derived. In the meridian section, Pig,67. through M: Iffl = N: IIG = n: MGD « geographia graphic latitude - L: W.CD = geocentric latitude = L, : B„ = radius of curvature of the meridian; x and y - coordinates; a and b • semi-axes. The equation of the ellipse is 31 V a* + y'-Zb* 1 or b^x* + a'-y* - a'-b^ (a) Differentiating, 2x dx/a* + 2y dy/b'' = or, dy/dx = -j b''/(y a» (b) Prom the differential triangle, Pig. 67, ' ^^7 6? HADIDS OF CUS7ATURE C%91,Pig.67, dy/dx -cot [i " -003 Usia L (c) SqaatiQg (a) and ( hj , b'-x/(3^7) cos Ii/sia L, or b^nVCa^y) " a* co3*L/( b''3in'D) (d) Prom the defioitioo of eooeotrioity, ^«. aX( j gM Substituting in Ca). xKl-e'-) + y'^ ^Hl-e"^) (e) Prom (d), x^d-e'^^sin^L -y'^oos'-L = f f ) V-altiply (e) by cos'^L and add to ( f ) , x'-tl-e'^) (oos"-L ♦ sin^L ^ e'-sin'-i,) " a'-Ca-e'-) cos'^t x'^^ a'-oos'-i./Cl-e'^sin'-O) (42) Multiply (e)' by (l-e*-)3in^L and subtract (f). y'-r a-^('l-eM'-sift'*'[./(l-8'-sin''[.) (43) Putting l-e'^siD^'L r'' X a cos L/r y - a(l-e*')3ia L/r (44) 90. FBINCIPAL RADII OP CSJRVATO8E. Since arcs subtending the same angle- are to each other as their radii, the radius of curvature of the neridian. 8^= ds/dL = -(il/sin L)(dx/d[.) Prom (44), dx/dL =(-ar sin L + ar"' e'-sia L oos'-D/r'- Substituting, -a(l-e-)3in L/f^ a„ a(l-e'^)/r^= a(l-e^)/(l-e'^sin'-L)^* Cmst) The. section by a plane through the normal MH andXto the meridian is called the prima vertical. It is tangent to the parallel of latitude at M and its center of motion, or of curyature.is on the axis' at H as the point M moves past the meridian plane, .'.from Fig. 67 and (44), Radius of curvature of prime vertical = normal ending at minor axis, H x/cos L a/r (46) Dividing (46) by (45), ll/a„ r'-Zd-e'-) 147) This ratio is often of vjQue as indicating the deviation of the sarfacei flt any point from that of a sphere. Por L = 0° ll/R„ 1.0067 L - 45» R/R^ 1.0034 15 1.0053 60 , 1.0017 30 1.0050 90 i.OOOO The geometrical mean of N and R^is taitea for the mean\adius of curva- ture at the point, i.e.. Mean radios of oorvatsre, g -J k Hy^ (48) Radius of parallel, 8,= » - a cos L/r 8 cos D ' (49> Normal ending at major axis , n = y/sin U " a(l-»e^)/r (SO) Geocentric latitude, Pig. 67 ,.thepoIe ^^° ^r r'^ =(l-e'-) tan t (51) Qi^t,, varies from 0° at the equator Eon'40" ia latitude 45°and 0' again at 91. RADIOS OP CORVATOKS FOR A GIVBN AZIMJTH. A plane through the normal H6 outs oat an ellipse. Its equation is found by expressing the coor- ;dinales of a point in the equation of ^^^ surface in terms of the co- Gq.54;) l^S&IH 0? MSBIOIAH ABC. QQ orainaiss ot the carve. PBe equation of the sarfaoe is For itie point P, ^ =0G+GR-NA=Ne''co3 t+yoos L-x oosz sin I y' * FN X sin z 2^ =RQtQ4 =ysin[i+x cos z cos L sabstitating in (a) and asiag for b, a VI - e*- x'^('l-e* ( l-oos'-z cos*[i))+ y'-(l-e*'0os''L)+ xy(2 e^'sin L cos L cos z)- 2x( l-e^)Ne^oos L sin L cos z+y 2e*( l-s^)N cos'-m l-e'-)Ca>-ll'-e'= 1 Prom (a) R^sia* z/A + R^oos* z/8^= 1 or 8-1 =W8/(9^sin'z + N oos'-z) . (53) Table V. is compitea froir (53) 92.LB!ieia OP IIBRIOIAN ASC. Since R^ changes slowly »ith L.for arcs of 1° to 2°. ds R_d C" sin 1" l.^"'') «-«- rtjiere iU"' is in seconds, and R-n, is for the middle latitude. For long arcs (54Kmust be integrated. |gbsiiiutiag the value ol B, ds ^(1 - e^jjl - e'sin^trib- By Pormula32l, ds = a(l-eM(l t(3/2)e*sin'-L+(15/8)e''3in"'l.+(35/ie)e'3in'L)d[. s = a(l-e'-)Jcd+(3/£)e'-3in'L+(15/3)e''sin''[.+(35/16)e'-sin't,)dL By Pormalas lO and 12], sin^L = (1/2)(1 - cos 2£.) e4 sin'' L By Poroula S) , 8J, SPBBBICAL SXCESS (? 94. Fig. 70, tl/3K3-4 cos 2L+ cos 4LJ sin'L siQ'-Lsin''L ll/3£H10-15oosSL+Soos4L-oo3«.J SabsliLuiing and puiiing l+l3/4Je'«45/e4)e''+(i75/256)e'- = 1.0051093 Log. = 0.0023133 l3/4;e^+U5/lS)e"'H525/51£Je'- • 0.0051202 = 7.709237 U5/64;e"'+ll05/25eje* =-0.0000103 =5.08342 136/512) e' - 2.326 s all-e'^) fiS^-^ cos 2L ♦ C oos 41,-0 cos 6L.,,Jdb •= &(l-e^){&b-{l/2)b siQ 2L+U/4)C sio 4L- (1/6)D sin 6£,...jlf Subsiiiuting the limits, and patting L'-L'^'^, L"'+L'=B,»e have by Poruula ■ 's=U. 4395369 ;=r»-( 4. 511036JsinofoosB+ll.534l4-)sin2'vco32l» -C8.651Jsin3o(-oos3P 155) Khere s is in meters and'ihe numbers in parentheses are the logs of the constant factors. SoaatioD (55) is correct for 7 decimal nlaoes. If acre are desired, the next ternr for A is I11025/1634)e»;for B,^g205/2043)e•;for C,l2205/409aj«*i for D,(315/£04S)e*,nhile an E term is adoed = 1315/16384)6* 93. ARSAS ON Id£ SLIjIPSOID. Dividingthe surface into frustrums of cones by the parallels: iiidtb = B^L; circumference = Z^H cos I. Differential area, dA = 2niJR„oos I ah la) Substituting for N and R„ from 145) and 146), Mitb b'' for a'^U-e'-). dA = 2Tib*cos L dL/U-e^sin'L)'' Bs fotmala 32], {. l-e^sin' h)"^ l+2e"-sin^ L+3e"'sin"' L+4e'' sin' t +5e»sin«L Substitutinfi.the expression to be integrated beooir.es _ _., Soos L sin"L dL = (lAn+l))siff*' L shich gives, ^c^ £bVlsin L + (2/3)e*sin* L + l3/5)e"'sin''l, +14/7)6* siiT Ltj|^, A^J 2bV(siD I"- sin L' + (£/3)e>(Bin''I," - sin'L') + (3/5)e-'(sin*'L"- sin*!.' ) + (4/7)e'(siqi''L" - sin"'[;)+ ) (66) To put in convenient form for computation, sin'L = (3/4)sin L - (l/4)sin 3L sin»L (5/3)sin l- - (5/16)sin 3L + (l/16)sin 5L siniL = (35/e4)sin L - (21/e4)sin 3L + (7/e4)sin 5L -(l/a4)sin 71. Substituting in (56). »e have by Formula 8],Bith L" - L' = SYand L"+L' =8S, a' = 4b''Tr(b sinv oosS- sin 3>cos SS+C sin ovoos 5f-S sin 7Vcos 75.J(57) »aere B=l+(l/£)e''-*(c/8)e*+(5/16)e'+(35/lE3)e»' =1.0034016 U)g=0. 0014743 C= (l/e)e'-+(5/16)e"*+(3/16)# +(35/19£)e*rO.00li3eS 7.05568 D- (3/S0)e"*+(l/ie)^+(5/64)e* =0.0000017 4.2304 ^ ll/ll£)e'+(5/256)e'*«0. 0000000 F= (5/2304)e*=O.O000p00 If (57) be divided by 360 and L" -L' ' 2v= I'.tie afea for 1° souare. t=(b*"-/SO)(B sin 30' cosJ-C sin 1°20' cos 3S+D sin 2»30' cos SS -a sin 3°30' cos 7S-.> (58) The values of B.C.etc, should be carried to more than 7 places for ac- curate results iiith the Clarke ellipsoio.altnoueh ihe above values are car* rieci as tar as as tne data Bill warrant when applied to the earth. _1 94, SPHiniCz-L ii>C(il3S. In Legendre's tneofem it is proved that in a spher- ical triangle Jihose sides are" snort coipared iiilh the I'adiafe R of the sphere and a plane triangle with sides of equal length the corresponding apgles aiUer by ibe same quantity BCuct is oas-Lhiciine sph?ripal e;ccfcss. Let A.B.e, = the angles of the spherical ana A'.iB',,C', those of the plane t.riaBgle:a.b,o.(n--ii;easiire)aod a'.b' .c'.,ihe corresponding siaes:aK = a .tK Fx« ■)». Sq.aO.) EPPECT 0? HSIGHT UPON BOR.ANGLSS. bS = b' : oR - 0' . ' Plane trian gle. By Formula 21], j. ^ .. „. ^ . . *— ■" COS A'=(b+o'-a'-n2bc:)tF'-/fl^; \.a.l By Form.l], sin'^A' ■=l-oos'-A' -((Sb'o'^-(b'-+o^-a*)')/M'^"'«-'")( •*"/«.•') or, sin'^A'=Ua'-b'-+2a''c'-+2b'-o^-a''- b"* -o'')/4b* o*" ^b/ Spherical triangle . By Pom. 27], u , ^ -*— — -Tj — -" — COS A-ioos a-oos b cos ol/fein b sin o Ey ?.l-dj and I4J . =(i.aV2+a''/ 24-(l-bV2+b''/ I4Ml-cV 2+oV24;) /U-bV6no-o5/6) =((-a^+b^ +0'^;/ 2-1 b'^+o'' -a-* J/ 24-b'-oV 4i)A''0^1-^ '>'■+='• V 6) -K-a'.+b'- +c>-;/ 2-( b"* +o''-a'< J/ 24-b'-oV 4) U+^lf■ +o'- J/ ti; Ybc =((b'-+o'^-a'-;/ 2+lb"'+o"'+2b'-c'' -a^ c'' -a*- b'-J/ 12)'/lrc. -(D''+o''-a'<+8b''oV124bo , U) From (a; and Kc). , „ ,,,c,>. • "K. ,j, oos A cos A - (1/6) bo sm A (a) Since b and o are very small, the difference betneen A and A' must oe small. Putting this difference ' x, . . „ „ oos X " 1, sin X X sm 1 cos A = cos lA'+x;, by Porm.4l, , „ „ . «• •= oos A -X sin Ism A; py Ui, .,,. , ,,, -=003 k'-{l/B)io sin*-* or X =■ DO sin A /6 sin 1 i.Q) nhere b and c are in ir-measure. If b and o are in units of length on tie SDbere of radios P. t ^ v •, ^ x" ■= bo sin A'/ 6 RSin l"=area of triangle/ SR'-sin l'' The same can be found, for B-B' and' C-0' . Since the areas of spherical triangles are to each other as th«.\T spheri-^ cal excesses, we hare from the trireotangular triangle,.excess- 9P°, Spharioal excess in seconds, s,"=area 2 90° 3600/wR'^=.5bo sin AVB^in 1" Comparing this with (a). Spherical excess, =,= 3x -.Sbo sin A'/R^sin 1" (o9i OT, s " m be sin A* (SO) where m = .5/R*sin 1".= .51IR,^in 1" by (48), -and is given in Table VI in letric units. 95. EPPECT OP BSIGHI OPOU aORIZONTAL ASGIiBS. Tlie observer at A. Pig.'71 ,at sea level sights upon li at the height h-cs-above at B , The vertical' Slane of oollimatlon at A Drojeotsw to B on the line drawn to Ha. where the normal at A meets the axis, while the true ptojection is at B', on the normal MH^ to the surface at U. This makea an error x in the horizontal angle at A due to the height h. Pirst to find the angle Be- tween the two projecting lines at U. In Pig. 72 let C be the intersection of the normals at U, and llj both in the same meridian. If As is small C jrill also bathe center-' of curvature for the arc M, U„ and Piq_"H. Pi<).1l. As- = a^AL If Ml be produced to meet the, axis at^H, .and the reduced difference (a) he (b) in latitude fi,H, U^ be called At' (U, H, = S) As = N4L' SroB (a) and (b), Ati/AL' = N/R m 6iit,S'=ikI, ,M,' =Ai:i(l -M'/hL) =&[, (1 - R„/H> Prom the values of R„ and N,(45) and (46), B„/H - (t - e*)/(l-eHin^[,) Substituting,S=A£, (l-((a - e'-)/(l- e'-sin* ti)) = i£i e»-003'-L/( 1- e'-sia»-Ij) from Pig.71iR^AL = - fc cos z,aearly (lat. = dist. x oos of bearing) GEODESY. =-k e''003'''t 003 z/(l-e''sin*£i) R (596, Pie. 78, =- lte^co3''£i cos z/(l-e'-)N BB' ~ hS = - h }c e^cos'^Ii oos z/ ('l>e'^)N Ihe oorresponding horizontal angle error at A in TT-msasure, X = - SB' sin- z/k = h e^'oos'^ti sin z oos z/Cl-e'') 8 x" h e^cos'-L sin z oos z/(l - 6!^)S sin 1" {61jt This ?(ili be a laxiiuir. for z - 45°. If U also 45°, x" .000055 h (62) where h is in meters. For a height of 1000'^ this would give 0."05. ihj orobaole error in the value' of a primary angle is seldom leas than ■O'/So, so that tne above cor- rection would be negligible except for very high altitudes. 96» TRIAHGL3 SIDi C0MFDIATI0S3. The triangles of a triangulation are strictly spheroidal, but by §95 the 3 vertices of a triangle can be pro- jected down to sea level by lines drawn to the center of a sphere tangept to the ellipsoid at ihe center of gravity of the triangle and having \/SS^ for radius, only affecting the horizontal angles within the limits of the errors of observation. The sides of these projected triangles have the same lengths, »ithin the errors of measurement, upon the tangent spheres as upon the ellipsoid. The triangles can thus be considered spherical, and by Legendre s theorei:. computed as plane by subtracting one-third the spherical excess from each spherical angle. In' simple systems, and where the greatest accuracy is not desired, if the sum of tiie observed angles in any triangle does not equal 180° + s,or the sum of those about a point 360°, the error is distributed equally amoDg the angles, or sometimes inversely as the number of reoetitions. But in oomolioated systems, or where extreme accuracy is desired, the er- rors are distributed oy least squares. The following is a convenient form for computations Base 0= 6410.6S ft. w e Ow 0^ Sin A'^ Sin 01 A = w e Sin Aj Sin 0^ A A 3.8069028 .... 9.9907935 3.8161093 9.8593280 3.6754373 4736.28 3.6754373 9.8135605 Sin O; . 3.8161093 9.9156182 3.7317275 A 5391.72 3.861876S 3.8618768 9.9999957 Sin A^ 9.3819486 3.8618725 7275.66 A 3.7438254 5544.03 FiCi-ns. CHAPTaa VIII. GBODSTIC POSITIONS. 97. DIPPBRaSOS OP LATITUDS. It 13 uaual to find the latitude aad the loa- gitnde of ooe or more of the triangulatioa stations by astronomical obser- vation, as also the azimuth of one or more of the sides, and from this data to compute the positions of the other sides. In Pig.''4, P' is the pole of the ellipsoid and P that of a tangent sphere. Ihe latitude of A and the azimuth z and distance K to 6 are given. Since k is always small, its subtending angle being usually ■ 8 ain*z cos z tan Ij(dz/dm)-3in'-z sec''[i(dL/dm) 2 sin'-z oos z tau^Ii + sin'-z cos z(l-+ tan'-Djjy (a)and(c) s sin'-z 003 a. (1 ■•■ 3 tan'-L) substituting in 33}, Ii'.-L = - m cos z •^ClnV2)sin*z tan LJsi'/8)sin^z cos zd-fStan'-L) (es) Where L'-O and m are in-wneasure. , For radius S, m = K/N and, [,/.[, = - (K/H)cos -^ -(HV2H'-)3in'-z tan h -+(kV6«>)sin>-z oos z( l-+3tan'- L) If the center of the sphere is- taken at Ho. it will be tangent to the ellip- soid at A SO' that h nill- be the same for both.aS' also- V and z. Ihe linear difference iff latitude Hill therefore be the same* for each surface, i.e., {■L*-L)N - AL sin I'.R,^, or M, =(ti'.-I.)H/R,., sin- 1* (e), where aL = differ- ence in latitude in seoonds for the ellipsoid, and B^ is for the middle lat- latitude. Substituting, -iiU - (X/B„3in- I'jooa z -+(>lVlHR„ sin 1") 3in>-z tan L -(ie/6fJ'^E^,sin l")" 3in»- z cos z(l+ 3 tan'-L) (64) It is inconvenient to look oat Km for the middle latitude which is at first unknown. If Ru is used the resulting difference in- latitude SO will be ohan:ged in inverse ratio- to the radins.by (e),i.e., Mj : SL :: tij R„ ot, 4L =fMB-/B J =SLI- 1 -(8. - R_)/R )=SUl-dfi /R ) L m *• *" m mm i.e., the true value can- be fonnd by subtracting ^Lda_ /B from the approi- value. ' „ "• m sin- z cos z lCV\ Prom (45), a(l kB )/n 2 . 2, ,3/2 e sin L)"' 89 G30033I. (S99,Pig,74, dR„ = all - e'-) 3 p.'-siti L ooS' L dL/U - e^siQ^LT' SiQoe dEi„i3 the change from the starting point to the middle latitude, dL/sin 1" 'SCi/a. :.SL da„/R„ - 3 e'-sint.oos ti sia r(i'L)V2ll - s'-sin'-G) Placing D 3 e^sin I. cos L sin 1"/£(1 - e sin L), The correctije term = (f L^D {6o) ^ If B = 1/R„sin 1" ; C = tan- L/SS R^sin 1"; h = 1st. term of (84). which redness the 3rd. to h lOsin'-zd + 3 tan''Xi)/6 W'' Uith B = (1 + 3 tan'-£i)/3N'-, (64) finallsr becomes, - AD = V S cos z + X'C-sin'-z + {SL )'-D - h k^E sin'z (S3) B,C,0 and £ are gi^en in fable IV, Lhe anit being the meter, ffor secondary triangnlation the 4th. term can usually be ojiitied. 93. DIP?aR5!i!lOs IS LONGIIuDji. By Formula S6, sin Ai! = sin- m sin- » /cos L' Seferring to a sphere tangent at B,iT-s center at H-^ ,2,[j'k and W, are the same as for the ellipsoid, »hile m = yS . ain-M = Vsin z/S'. cos W (67) It is more convenient uO assame 4M - 4K sin z/ cos L' (63) ihere 4 ■- 1/H'sin 1", and correct for the difference bstireen' arc and sine. Formula 13]_ sin x x - f/6 ...■ = x (1- zVe) fformula 33, log x - log sin x = 1! x"-/6, uhere .¥. modulas of the common system of logs. logdog X - log sin X) = log(M V^/3 sin'' 1") = 8.3303 -f 3 log- x for AM, logdog difference) =8.2-303 + 3 log Aii" (69) For m = i/H'sih- 1", using an" average value(8.5b90) for log -1/N'. sin 1" or log 4, logdog difference) = 8,2308 i- 2 log k •^ 2 log A= 3.3433 ■^ 2 log k (70) Placing ,8.2308 + 2 log ASS" = 5.2433 -f 2 log K log \- log ^M" = 1.4910 for the same log difference (71) The correction- for log K is and for log AM = +. The values ara given- in- -Tatle VIII. 99. COSVaiEiGSflCH; OP MSRIOIAilS. Formula 23]^ tan (A + B)/2 - • cot (0/-2) cos (( a - b)/2) /oos% ■^b)/2)) Substituting, ffig. 74, cot(Az/2^= oot{AM/2)cosjlL - i:.0/2)/(sin<(r. +L'J/2)) or, tan-(Az/2)= tan(Ayys)siQ^(L +L'J/2)/(cosa i:.')/2)) (72) Formulas ifl and l51, Az/2 = tan(Az/4-(l/3)iad>(Az/2);tan(AM/2)=(AM /2)+ C lt!/z}/Z substituting in' (72), +^z = AM sin D^/oos AL-f (2/3)( AM/2? (sin L^oos AL - sinl^oos'AL) or ifitl^ Az and AM in seconds, »"ita cos AL 1 in the corrective terms, -i^z" =AU" sin L /oosAI, +(1/13)( AM")" sin i, cos'-t sin»- l""l = AH" sin t.„/oos At, -f (Arf F J where F = (l?'l2)3in- [j^ oos'^ Ij^ sin'- 1" tabulated in Table IV. The inverse azimuth, z' = 180° ■!■ z - Az (74) For forms- of comoutabion see U.S.C. & 3. Report 1894 d. 237. lhe adjusted spherical angles must be taken and not tne olane ones used in computing the triangle, sides. For each triangle, starting from the knOHQ side, the latitude and longitude of the required point must be the same com- puted from each of the two sides, while the inverse azimuths of these two sliss aiist differ bi the third anele.thas cneoking the workt Bd.?^. ) OOOATION OF GREAT ARCS, 69 100. POLIfCONli; ilAP PBOJBOIIOH, This ppojeotion ia the oae most gener- ally used la platting geodetio and tapograptilc sarvesrst It supposes each parallel of latitude to be developed apon its own cone, the 7ertex or uhioa ia on the axis at its- intersection with the tangent to the meridian at the parallel. the side of the tangent cone, or radius of the developed parallel, Fig. 75, r = » cot L l^S) If an arc of the parallel subtend the an- §le Ml before deyelopment.and'after derelop- ent, 9 = A.M R,/r = AM S cos L/M cot L = AM sin' h (73) The radii of the developed parallels are so great that the parallels- are plotted by ooor"i«^ J"- ) din&tes. X r sin e = S cot Ii 3in(All sinL) C-n) f^t ns. y-,= X tan- 9/2 = x tanUin IiAU/2) IB platting, a central aerldlan is- drawn- as a straight line npoa- the map, -and the true distances between- parallels' are laid off from Table «K, Per- pendiculars-, by descriomg arcs- uith a compass-, are carefully drairn-, through these points for the )6axe3- of the parallels. The x' coordinates- are then laid off on each ror the different longitudes- (77) from the Table . Perpendiculars- are drawn- Lhrough these points- and the y coordinates laid off from the Table. The meridians- join the points- of the same lougi - tude.and the parallels those of the same latitude. A glance at Pig. 75 will show that, starting from the pole where the ra - dius- of the developed parallel is zero, the radius' increases- more rapidly than the distance from the pole, becoming infinity at- the equator; the de- veloped parallels will then not be concentric circles- but the distances- be- tween them will increase with the longitude from the central meridian-; distances- in- latitude will then be stretched out as we leave the central meridian, distorting the map since the longitude scale is constant. The triangulatiOQ stations- must then be plotted by latitude and longi- tude .interpolating between" the nearest meridians- and parallels-,and using the triangle sides- for checks- only. 101. -.aacSlOR MP PaOJaCTION. -rhis projection- is used by navigators- on- account of the facility in obtaining directions- for constant bearing sail- ing. 4 tangent cylinder is- drawn at the equator; the meridional planea- are produced to meet the cylinder in elements-.and the cylinder is- then- de- veloped. The meridians- thus become parallel straight lines- at dis- - tances' apart equal to the trua distances- at the equator. This- enlarges- the scale in- longitude in the ratio a/R-p . lo preserve local hearings- the latitude scale mast be increased in the same patio; the loxo- drome or curve of constant bearing at sea thus becomes a straight line nXih the same bearing -on the map. To find a sailing course between any two points, the navigator joins then with a straight line on the map, measures- the angle made with a meridian- and allows- for the magnetic variation-. In the differential triangles- EOP ,lcp, dm/ds = LP/lp - ee'/lp = a/E, Substituting for ds = R„ dL and for R,= H cos- L dm (iIVn cos l')dli (77) Substituting the lalues- of R„and H and integrating between the lim- its- L, and Ii,wiH give the distance on the map between- the corres-pondinfi parallels. " f^. 102. LOCATION OP GRSAT ARCS. Jf the two extremities of the line are given- fhe latitude and longitude of each is accurately determined by observationv' The azimuth and length of the line can then be found by (64) and (53) re - taining only- tw? terms of (64) thus, AL - V. cos z /R„sin l"-\Osin''ztan- L/2MR^sin 1"; AM =k3in- z/S'cosij'sinI'' solving for Vs-in- z, k sin z = AMS' cos L' sin 1" substituting for Vsin-*- z and solving for V cos z, Vcos- z = -r R^AIi sin 1" - M'^'AM'^tan- h cos"- L sin* I'VN 7. cot z = -R^AL/B'^AM cos W - AM' tan L cos W sin l'/2N \ a»,. ,.14. X= N' AM 003--'D'. sin- r/sin z ; (73) TO GSODESlf. (S103,Pig.77, If Ti is large it may be necessary to employ sei^eral triangles in locatla^ It or ;o test the direction by an obserred azimutb at an intermediate point. ioa.LOCAIION OP PAiiAij£jEljS. Sirst to find any point A of the parallel, a station A' as near the parallel as may be is. occupied and its latitude de- termihed; the difference between it and that of the parallel giyes the o^t dL to move either north or sotLth on the meridian to reach the parallel, or in distance at sea leyel, \!.= R-dL'sitt 1" (19) tan m ' m X (Placing • If dL is large the latitude of A sttould be deter- mined by a ne» set of observations on account of ihe danger of station error. Having one point A ,the parallel can be deter- TBined by offsets from the prime vertical AB. tn tne^PAB, Jormula ^s). tan Ml cos b. Formulas' inland IS^, tan- AM cos l -U/3)tao? AM cos> [> =M« oos L+U/3)( ttM cosCiJtan^ V mM - B sm 1" ^K" cos b +U/3)N (sin l"AMoos. bf tan* li (80) 'z = 90° in (fl ) for the prime vertical, -&l>" = V^ tan USS R™sin 1" 60 =.-iL''S„sin 1" = "t^zaa U/ZS Since BC varies as if if AB, or k,be divided into to yie parallel will be ( 1/n f BO, ( 2/n )^ BC, ( 3/n )" BC, ■ ■■ • ■Ihe direction angle, PEA =90° - t^z (SI) equal parts, the otai^^ws (82) (83) shile those of the n - 1 ordinates, assuming k to increase prouortionately to AM, Bill be. (84) '90° - Az/n, 90° 2Az/n, "0° -3Az/n--- If the parallel to be located is long &M should be divided into sections, and each one located from a ne/r crime vertical to avoid long offsets. Errors of direction- may be prevented from accumulating, and station errors may be detected , by observations for azimuth and latitude at the begin - ning of each new prime vertical. Ih locating the 49tli. parallel nest of the Lake of the floods, (U.S. Northern Boundary Survey, ifashington,_1878 jastronomioal observations, for latitude and azimu"^,ii.ere taken at points" about SO miles apart, and the prime ver- ticals were ranged through with transits. Saoh offset ifas. made up of ; the redaction from the prime vertical tc the parallel, increasing as the square of the distance from the astronomical station to the parallel, con- stant between stations; the difference hetneet the observed, and computed latitude of the closing point made up of the station and observing errors in latitude and azimuth, and the aligning error, and taken proportional to the distance. The probable error in the position- of a latitude station was about 4 feet, and in prolongong a EO-mile line, about 10 seconds. Example 1. Required tha data for locating the 42nd. parallel between- N.i. and Pa. from the Delanare River (apprgx. longitude 1° 30', S) to the west end of the state (approx. longitude 2 54 if) total distance 4° 24' lon- gitude. Dividing into three equal parts, we have AU 5330" = Mi Log sin 1' cos L 1st term=121517.S k* tan L 2N OS - 1040. 9 »« 4.6355749 3.7226339 9.S7 10735 8. 2792823 6.8053577 5.0346400 10. 1893648 9.9544374 10. 1233022 7. 1063377 3.0174145 U = 42° log(sin 1° 23' r AM 008 L)» N 1/3 tan'^L 2nd te: mslh 4.83785 6. 80536 9.52233 9.90337 1.07496 k = 1215297? ■n^«.« ti.'i sin Li 3533". - 53' 53" 90° 39° 01' -07" 3.72263 9.32557 3. 54320 3q.94. ) R3CT4HG0LAR 3PH3RICAL COOHDIKATSS. ' 71 foe ordiaaues aod direction angles for intermediate points can be foimd by ('92) and (34). ' 104. PAHALbStiS er SOLAS COaPASS. If a [i = in (T8 ) Got z = -(1/2^AM tan ti cos h sin 1"; X^fiM oo* t. siir 1" 3ub3titating, cot z = - V tan [,/2H =(az/2) lan 1" (85) 'fne first mstruneat point 'being apon tHe parallel, tHe solar »ill. gi7e the csridian.froi .vliich z can be turned off and the next instPumentVplaoed ap- 3a iri2 pajjallsl; etc, y'mt rne aifier^aoe in length, d.betireen- the north and south lineS' of a toitn-: ship «ill be the distance V between them into the convergence in- seconds', tines tan l • i ~ \' dz tan 1" (68; J'or long distances the difference should be found by computing the arc of ihe parallel for each latitude and subtracting. 105. BBOT&HGDLAR SPESRICAL OOOaDISAISS. In- Europe the positions of tri- angalation poioV nasre been found more convenient for use by local survey- ors when exprebded as coordinates' than as latitudes' and longitudes. In tae rectangular system the merid- ian for tae survey is dra/in through the origin and a great circle X to it through the required point A. ■Phe coordinates of A are x and y.and of B, x' and y' , positive to the north and east. '/ The bearing or direction- angle oc iS' the angle made, k,. not (lith the meridian through A, but »ith the arc AP p parallel «ith the initial meridian (the parallel' arc \ AP being X to t.ie great circle through the poles QQ'J. ■la find tne ooordlnaces anij direction aogls ai S from taose ai A. In the triangle A B Q tae 3 sides . are kno'.in as also ihe angles at Q('= (z'-x)/R) — -\- — pC^a, and A( = 90 -of). S ^"(-t^- :. for y', ?orm. 37], cos B9 - ooS' AS cos AS + sin AB sin AQ ons A /.„ sin>C+ y(-kV2S'-- yV6BS■^ k sino<(F«'**-'*/ti«.») y' = y ■^ k 3in«-(3 k'-y - 3 k^'y sin«e< + k* sinof- k'sin»o< )/8R^' y' = y -t- k sino/2R'')= k cos « ( l-W/breV-iVxil^ For a first approximation, x'- x k cos « substituting, I'.-x = (k cos«)'/3R'- + k coso<- k'cosor/ea'' + K y'^ C03V/2R"- or, x' = X -<■ k c03o< ■^ ky'V cosor/2a'- - k'coso) k k„(l - (cos'or/5R'-)(y>- + yy' + ?'.»■) (95) where k. is the value for plane oooriinates. Putting the map magnification' = S, e k./k 1 + (r* + yy' + 7'^ )oo3V/3R'' (98) ffor short lines y = y' nearly, giving G = 1 + y"- oosV/S R*- (97) This becomes unity for,3in'>'L) = q»- .•; e«-= (a - q*)/(sin»>L*- q^sin'-La (99) Since R„= a(l 3'-')/(il - e^3in^hf'\ R„= c((l e*^(l - e'^sin'-ttl Substituting in (98),, j, ,. ,., i, ^ c = IC'l - e'^sin^D'VAL sin l"Cl - e'-f*' \ (100) c 3- (a e-'sinVL'fA/AL' sin l"(l - s*)*/* J Semi-miiior axis, b ~ c(-l - e*)J Semi-major axis, a eVl - e* (101) The entire quadrant can be f.oana from (55) if desired. 108. REOrCTION OF A MEASURED ARC TO T33 l^ilSIDlXs. The arc is sup- posed to make onlv a small angle with the meridian. From (64),.. ' '3 = flL H„3in r = -k 00s z - (k'-sin'-z tan L)/a? + k»3i.n''Z 00s z (1 + 3 tan*j:)/6N'- (102) The second term of the second member is small so fhat an- apEroximate value can be used for S. For' a chain of triangles, this equation can' be applied, to side after side until the jrhole length of the chain 'lias been projected. 109. TH3 saiDIAN PROM SE/.SRAL LATITUDE DEGREE MSASOEElfflNIS. This inr volves the formation of observation equations between the observed lati- tudes and the pi?o jected, or directly measured, meridional arcs. The sim- plest relation is (93) which can; be used for a AL of sever.al degrees on account of the probable error of a latitude determination, some 3.04" or 4 feet, aside from the station error. or longer aggs a oorzisctinn for C^S) vrill be reqaired. ^xaHa. .■Bq..i04.r MESIDIAH BTOM DSaRES agiSOHBlElHS. 73 Prom (54), ds. = R_dL = a(4 - e»rd[/(.i - e»3in*L5^». = a(l - s*)dL(a + (g/2)eisin«'L). neglecting terms abo7e e*" ^ flit sin^L - 1/2 - (1/2)003 2C. ds = aC'l - e«')dLf'l + (3/4)e'- - (3/4)e»oos 2L> 3 =-,a(lr.7 ^((1 + (3/4)e^)(L"- L' ) - (3/3)e»(ain 2lf -sin 2L) Form. D , 3'= a(4 - €»-)Cl + (e/4ye»)A[. - (*/4)e»-sin AC cos 2L) for sin AL ase AEr - (ALf/3, s = aMid - e'-)(l + (3/4)e'-- («/4)e'-oos SL <■ (■l/8)e>-(AL)' COS. SL) (103) Expanding 8., R™= a(l - e'')(.-l- + (e/2)e''a'in*L) = a(l - e»)(il + tS/4)e«-- (S/4)e>'0os 2L) ,■ the approximate 7alae bj- (08) for s, , •sj = AIj-R„= a4m - e*)('l + (a/4)ef'- (Q/4)e^oos 2L) Sabtraoting this- from the true value (il03.)' irill give the correction' Ss to' apply to $be approximate value, or 3' - Si =jrs' =a ftL(l - e*-)( irtiere 1/8^ ■■ (1 - e*sin'L5'ya(4 - e"') ^b) Since a and e* are aninioirji.or required quantities, »e subatitatefor titea approximate vSlues irith oorreotiona, a = a + f a e'*' = e> + S* and expand by Maolaurin's theorem * 1/R„=1/H. +(d(l/aJ/d(«a))S-a +.(d(l/Rj/d(fe»))fe* (w) But d(il/Rj/dfo =((1 - e»'3in»Lf''V(i - e''))(-l/aj). 1/ a* by neg- lecting all terms containing e' d(dyRJ/d(«ev) = (d("/aj/dfe)(d(i-e)/d(-ti'vJ^C>-e.H''xe. = ('l/a K'l - (a/2)sin*[j), by neglecting e*- terms. SubstitBting in (O, l7t„ = \lK- ('l/ai)fa + (1 - (■3/2)3in^L)S^B^/3. .■,* {"«) becomes- Ii^ = L. + s/R,sin 1' - s('K'3^3in r>fa s(l -(3/2)3in''L)Se»/a.sin 1' -fs/^^iin," The value of is is given in (104). Considering the meridional arcs perfect or constants in comparison with the. observed latitudes, irith corrections v, affected by station errors the 74 e:0D3Sr. (SllO.Fig.SO, observation equations, (£1) Part I.beoome.iflth t., + V, = V,; L» + v, = Vi V, - L, = V, Vi - L^ = 7^ se BdtVi =V, + 3/R.sin 1" - sd/ajsin' l")Sa + s('l.- (•3/2)siri' L)('l/a.3ia X% Sabstituting, -S»/R,»iTM" 7. + L.- L + s/R.sin 1"'- sd/aisin I'iSa + s(a - (3/f2)3in'-L)(l/a^inl")5e'' I.e.. 7, + a^x <■ b^y + 1^ = 7» wiiere^lOOO s{l/a^siii 1') = a^; (fa/1000 = x; 9{1 - (■3/2)sin»L)(il/1000 = 0.003 6744 + 0.000 2347 = 0.003 9091. SabstitUuing the 7alu.es of. x and y in the obseryation eouations is) t,he y's are readily found, fr om ifhich fy-l =-52. ;. (31), Part I, £ =Vt7g/ln-mJ = >/527l2 = 2."1 for the m.3.=.of a lat- itude dstermiaation referred to the ellipsoid. This is yery much great er than the m.s.e. of a latitude determination showing that an ellipsoid of reyolution .Till not fit the data without large station errors or local deyiations of 'the plumb line. The 7's for each group, i.e., French, Sn'glish, etc. foot up zero within OiOl. 110. TaS ELLIPSOID .PSOV A D3(S33 MSASURBMSST. 0BLIQ,U3 TO Tag MSRIDIAN. The latitude aniJo azimuth are obser7ed at each end of the line, as also the difference in longitude and the distance. Saoh obseriration would giye an equation of the form f('X,Y,Z, ) - M,= 7, where the recpired quantities are the most probable values for the obser7ed L,,L .'^M.Z, ,Z ^»2°'*,P ^""5 ^"' fgt" ^s elliisoid. Denoting the odrreotions to* the ob-''' aeMed or assumed values byj.we ha7e for the initial latitude f.(t, + S[,,) -Li= 7, . or SL, '0=t,' (a) «lq.lD6.) DSGa'Dal MSAS'jaBMSH'JS. 75 For L,. f,.(Ij,+ SL, ,o.+So,e;' + Se'S - L^= Vv . ^^ , >^ ""i The quantity- fvC'^i ,c«.ei') .the compated valae of L^,oan be found by (64). Place this oompited valae less ti^ ; !», ^ 1°^ the.differential ooeffloients only the first term of the second member of (64) need be used. I.e. , f^(G, + STi, ,o.+ro,e'.*+Se;^) = L, - k oos z/R„ = [i, - k oos z.VVc (idf^/dL, jrL, =yL, (iJfi/do )So =(k oos V?/c^Jt;(dfj/de-MSe'M-3k/8o) oos zx Cos.'E.V.Se'- Collecting results, SL, + (* oos z,V>'/o^)fo - (13I5/20. )003 z, oos^L, )Se-*-+ 1^= 7^ (c) For an, ' Place 1,= computed value by (68) less the observed value, irhile ■for the differential formula use k sin z/Nl 003 L^, i.e. , fj(c„T S'o,e;'+ Se;*-) = u 3in z/N'.oos W. = k sin z.VlVo oos l^ (idfj/do)ro -(k sin z.V.'/Coos L^)ffo ('df,/de'MSe-'-= (k sin z, oos'Ci^A .•. -('k sin z.V'/oJoos L^jSo + ('k sin z.oos'L^^o.VOSe-' + 1,= v, (d) For azimuth, f^('z,+ Jz, ) - z,= v.,, or fz, +0 = v., (e) For Zj, tj.'z, + Sz, .0.+ X"o,e;* + Sef^) - z^ = v,- fs,.(' ) by (73) z, + 180"""+ m siir Ii„* z, + 180° + k sin z.tan L/n' = z, + 180° + k sin z,tanti,vyq (df4./dz,)S'z, = ^z, (idf,/do)jo (-k sin z.tah- L.V/cSfo (■df^/de'»)Se'* j: ('k sin z,3in L.cos L/acvOS**- .". Sz, — ('k »in z, tan- L.V'/oj)^© + (* sin z^sin L.oos L^v')Se'''+ lo-= v, IS) Ckslleoting equations (a) to (f) and denoting the coefficients of So and ye'^by a and b, SL, 1, = ^• *L, a^fo + bje'* + 1^ = Vv a,ib + bjle"- + 1, v, (106) Sz, 1^ v^ rz, aj-yo + byje'» + 1,. v». Weights can be introduoed if desired. If k is large or poorly measured so that its m.s.e.is appreciable in comparison- with those for L.AM.and z.another' equation should be added ft(o.+ Sc.e;'- + fe'») - k = V4 Prom (78), fc(i ) = N'AM oos L^/sin a, = o oos L^iM/V.'. sin z, (idf4/dc)rc = (AM cos ,I.,/.V,'sin z,)So (d^/de")* e"-= -(c AK oos%/ 2V'* sin z,)5e"- ^ . a^sc + b^ae'^t Ij = Vj is the equation to be added to (105). ifTa second line starts from the initial station and its azimuth is com- puted from the observations -.Thioh gave z, ,tl)ere ?f0uld be added to (103) ill, + a.rc + b,Se'^ + 1-, = v, from L, a,So t bjie*^ + 1, = V, °; ^M,.j Sz, * a,so + ll,Se'V + X, = t, a,.sc + b„ e'» + 1,. = v„ Ihe distance k can be greater than a tri- angle side by- solving for an approidmate z by by (TO); computing through the ofiain of tri- angles with ti?o angles and the included side giyeh each time to find the third angle and the second side; calling the change in direc- tion of k at each intersection 180°.' z,, Zv 3hd k as found for the total distance can then be corrected for the error in closure *°i' at B by adding x to k and dividing y by k sinl" for the correction to z. For the more general treatment for an astronomical geodetic net, taking ito account station error in its effect upon latitude, longitude and az- imath.see Uelmerf'S Hd'heren -Geodasie. 76 ■TJELE I. Popmalaa andl Constants. sin*'x + ooa'-x - 1 Q tan X = 1/cot X = 3ia z^cos x =V3eo'x - 1 2) sia('x ± j) = sin- x cos y ± oos x sin y 3 cos (ix ± y) = cos X cos y zf sin x sin y 41 tan (ix ± y) = (tan x *. tan y)/('l:?: tan x tan y) Si cosC'lSO'*- t) - - oos y: sin('180''+ y> = - sin. y- C Pop small angles, sin x = ta^- x = x' sin 1" = sf'. arc 1' 7l sin X t sin y = 2 3in(('x ■x.\)/Z) cosCd:? y)/2) tf) cos(ix + y) + oos ('X - y) 2 cos x cos y 9^ sin- 2x = 2 sinr x oos x ' lOJ 2 oos'-x/2 = 1 + oos X l5 2 3iit>-s/Z = 1 - oos X 12} sin X = X - xV3! + x'VS! - xV7I + 2f/9I - IQ cos X = 1 - :^2! + x''/4! - x«/3I + x«/8I 14^ tan X = X -f x»/3 + Sx'VlS + 17x''/315 + 32x9/2335 + 1332x"/l55925 IS) are sm x = x + x'/3! + 3xV40 + Sx-i/lia + 35x»/ll58 + 63x"/SS13 1© arc tan x x - ^3 + xVo - xV7 + xV9 - x"/ll + ig In the last 5 equations x is in -ir-measure. Should x be giveri in sec- onds multiply by sin- 1" Plane Oblique Triangles, sin A / a = sin B / b = sin C / o ig) a*- = b"- + o'- - 2 b c cos A jdj tan((.A - B)/2) =((a - b)/(a + b)) tan ((A + B)/2) 231 Area triangle = 1/2 b o sin A 2fl Spherical Obliqne 'Triangles. sin A / sin a = sin B / sin b = sin C / sin c 23\ cot B = (Bin- c cot b - cos c cos A)/3in A gSQ cos a = cos b 003 ' o + sin b sin o cos A gA tan((d ♦ B)/2) = cotC!/2)oo3«a b)/2)yoos (.x«/nr) 33) Taylor's Theorem. a' - f(ix -^ y) = u -f (da/dx)(y/l!) + UWdx'>(yV2I) -^ (id'"a/dx'')/(r'/n!) 34l Badius of CorraturB. R = - (1 + dy^/dx'MdxVd'y)' 35J A.B.Clarke of the-Bnglish Ordnance Sanrey, gives the following yalires. for the ellipsoid of rerolntion as found froTii the various degree meas- urements. These values were adopted by the (I.S.C.& Geodetic Sarvey in 1875 and the follotfiag tables irtiich involvs the ellipsoid are based ap- on this data. "Semi-major axis-, a = 8378338."' 4 log a.804r:6985 seiri-minor axis b = 6358583.8 "• 8.803 2238 eocentrloity sqaaredp»-= 0.003738353 7.830- 5023 One meter = 39.37 inohes(iAct of Congress). The following formulas are in use. e=(a»- tf^/a e ' = (a*- .- B^Vb a = cVl - e''= c/Vl + e"- b = otl - e*-) = c/(>l + e'*> r'-= 1 + e'-sin'-L V>- 1 ♦ ff'-cos^t = ryCl ^ ^ The following approximate values are given for the coefficients of es^ pansion for 1° P.the unit being 1/1 000 OOO of the length. Glass- 4.7 Iron 3.5 Platinum 4.8 Brass 10.2 Steel 3.2 Zinc -K.l >■ ^ 4 -r^ ~ ' J Table (LCorrections for run of tt-ie micromefer Cerredions sqttic s'ljnos r for ni ti. $iiJi- Eg OS 10 .IS to .25 JO .iS .40 .■♦s .50 5S .£o ■£6 .Ifi) .15 .80 .85 .90 .95 1.00 1.05 1.10 1.15 I.IO WO /.»! 135 lit .05 09 ./4 .19 ti M a 37 .41 .47 .51 .56 6/ .65 .70 .75 .79 .64 .89 .93 9« \n 1.07 /.1 2 )./7 ao" 140' I so- .04 09 ./3 .17 .22 itG .30 .35 .39 Ah .46 .5X 55 £0 .65 59 .74 78 62 .87 .91 .95 1.00 /.04 1.08 6a" I SO" ,04 .08 ■n 6 .:io .M M .32 3( 40 .44 /(8 .52 .56 60 .64 .68 ■76 .SO .84 .88 .92 95 /./3 IM 07 .11 .15 .18 .n .» .a? .33 .37 .AO .44 48 SI .55 .59 .6Z 66 .70 .73 .77 .SI 84 .88 .92 .95 .99 |.«3 lx)6 (.10 03. .07 ■ 10 .13 •n .M .13 27 30 .33 J7 .40 .43 .47 .50 .53 .57 60 .63 •67 .70 73 77 80 .83 .87 .90 .93 .97 1.00 7o^ < g,'-ao". Opposite signs foK m ? a.''ao .03 .06 .09 .It .15 18 il 24 .27 .30 .33 .36 .39 ,42 /45 .48 .51 54 .57 .50 .63 66 69 .72 .75 .78 81 .84 87 90 n'4' .03 .05 08 .11 .13 ./6 ,19 ,21 ,24 ■27 .29 32 .35 .37 .40 .43 .45 W8 ,51 .53 .56 .59 61 .64 .67 .69 ■72 .75 77 .60 3°: iSL 02 05 01 .09 .1% .14 16 /9 21 .23 26 .28 .3o .33 35 .37 •40 -42 .44 ■47 .49 .51 .54 56 £6 .61 .52 63 .54 02 .04 06 08 .10 .12 14 16 18 20 .22 24 lb .20 .30 .32 •34 .36 38 Ae .4« .44 .46 .48 50 60"! 5o"Ho"l30"l ao"l lO .02 .03 .05 ■07 .08 .10 .12 13 .15 .17 .18 ,20 ,22 23 .25 .27 .28 .30 ,32 .33 35 .31 .38 .40 .42 43 .45 .47 48 ■50 Js. .01 .03 .04 .05 .07 .08 .09 II .12 .13 .15 .16 .17 .19 .20 ,21 ■ii .24 .25 27 28 .29 .31 32 ,33 .35 .36 .37 39 .40 «■«.' 01 j02 .03 .04 05 .06 J)7 .06 .09 ■10 .11 .12 .13 14 IS 16 17 .18 19 .20 .21 .22 .23 .24 25 ■26 .27 .28 .29 .30 .01 .01 .02. .03 .03 .04 .04 05 ■06 m 07 ■08 .09 .09 .10 II II .12 .13 .15 14 ,15 15 .16 .17 •'7 .16 .19 .19 .00 .01 .01 ■01 .01 .02 .02 .03 .03 .Od 04 .04 04 .05 .05 -OS' .06 .06 .06 •07 .07 .07 .08 .08 .08 .09 .09 .09 .10 ■10 60" I 50*140' .00 .00 .00 .00 .00 .00 .00 .00 00 .00 .00 .00 .00 .00 00 OO .00 .00 .00 x)o .00 .00 .00 .00 .00 .00 .00 00 .00 .00 o-=r Table Ml. cMeti-ic Unit's) nr ooo 30 I 00 30 2.00 30 300 30 ■loo 30 500 do £00 AO tot io eoo io 900 30 HOC 30 II 00 30 iVBai-gl. iii3:ii 1316 1304 1283 1253 1215 1169 1114 1051 09SO IIO900 OBIT. 07/S 0610 0497 0375 0245 0106 9959 9804 100641 9469 92B9 9101 rofMeni). /I0S67.2 567.6 568.6 570.3 57 J. T 1 10575:8 S79.5 583.9 £89.0 594.7 HtiOl I 608.1 Loq.V 30 696B 0552 8/90 £300 067S 'J64.4 6665 2590 30110 96488 / 10 846.5 6.805066 S 68026522 30 49806 6.8058769 e.8051901 ao 6ooJ 0775 8858 6400 8934 1/1480.3 689/ 3106 3100 5506 665.7 0888 9/97 30 6057 6992 35/0 '30 5004 /oa2 95je 6500 7/77 495.7 9091 3809 , 3100 4495 aea.z 1111 9683 30 6294 9/90 4/03 30 3979 /X33 58030 23/ 6600 5407 5/ 0.7 9287 4J93 3500 3455 901/ /350 0582. 30 45/6 9382 4676 30 X925 /467 0936 6700 3612 .5253 947 5 4959 34 00 Z367 9/9.1 /586 /2.92. 30 2724 9567 5235 30 161% /706 /65/ 68 OO /623 539.3 9658 5506 3500 9/290 //0937.6 6.805/826 660320/2 30 409(9 6.8059147 6.8055713 30 073/ /947 2376 6900 00/2 ///552.9 9834 6034 3«oo 0166 956.x 2069 274/ 30 39102. 99/9 6191 30 fl9S93 2/92 3/09 70 00 6/66 565.9 6.8060003 6542 3700 90/4 975./ 23/4- 3479 30 7272 0085 6789 30 8420 2439 385/ 7100 6 353 578.4 0/65 7029 3800 7835 994/ 3S-t1 4224 30 Stil 0244 7265; 30 7235 2689 4599 7X00 4506 590.4 031/ 749.5 3900 Sfi29 /l/0/3.^ 28/4 4976 30 3576 039 5 77/9 30 6fI6 2940 5354 7300 2646 60/.8 0468 7936 4000 85396 ///032.7 6.8053O67 £.8035734 30 317/6 £6060539 £.8058/52 30 4770 6 I.0015G 2.22 54 5.7780 22 00 fl.SOa 5802 8.5(Z05TI 1. 01254 2.2343 5-7651 7.600 30 8.509 5112 8.5120301 1.02341 2.2411 .5.7924 24 00 6.S09 5o2o 8.5(20026 1.04490 2.2485 5.7997 78(2 30 8.509 4927 8.SI/9747 (.04431 2.2557 5.8071 24 00 8.509 483) 6.5(19463. 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