EXCHANGE SECULAR PERTURBATIONS ARISING FROM THE ACTION OF JUPITER ON MARS A THESIS PRESENTED TO THE FACULTY OP PHILOSOPHY OF THE UNIVERSITY OF PENNSYLVANIA BY ARTHUR BERTRAM TURNER IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE DOCTOR OF PHILOSOPHY PHILADELPHIA 1902 SECULAR PERTURBATIONS ARISING FROM THE ACTION OF JUPITER ON MARS A THESIS PRESENTED TO THE FACULTY OF PHILOSOPHY or THE UNIVERSITY OF PENNSYLVANIA ARTHUR BERTRAM TURNER IN PARTIAL FULFILMENT OF THE KEQUIREMENTS FOB THE DEGREE DOCTOR OF PHILOSOPHY PHILADELPHIA 1902 EXCHANGE PRESS OF THE NEW ERA PRINTING COMPANX, LANCASTER, PA. 55 * ACKNOWLEDGMENT. I wish to thank Professor Charles L. Doolittle and Mr. Eric Doolittle for their generous instruction, and their helpful advice in the prosecution of this Thesis. CONTENTS. I. Lagrange's Generalized Equations of Motion. Lagrange's Canonical Equations ........ 5 II. Canonical Forms of Hamilton 8 III. Method of Jacobi and its Application to Two Bodies. Ca- nonical Constants . . 11 IV. Variation of the Canonical Constants and Jacobi' s Equation . 17 V. Differentiation of the Equations containing the Canonical Constants . 20 VI. Transformation of the Equations Expressing the Perturba- tions, and the Values of the Variations . . . .23 VII. Dr. G. W. Hill's First Modification of Gauss's Method . . 27 VIII. Computation- Action of Jupiter on Mars . . . .30 Biographical .......... 36 I. LAGRANGE'S GENERALIZED EQUATIONS OF MOTION. LAGRANGE'S CANONICAL EQUATIONS. Let F n , F 12 , F 13 , -, F ln be the forces acting on a unit of mass ra x , F 2l , F 2 F 2 , F 2 F 3 , - , F 2n be the forces acting on a unit of mass m 2 , etc. etc. Let 8p u , &p l2 , Sp 13 , , 8p lH be the virtual velocities of m l , tyPn. > ^22 > ^23 ' ' ' ' ^Pa ^ e *^ e virtual velocities of 77i 2 , etc. etc. Now assume that each mass m. be displaced an infinitesimal dis- tance I = ds. in the direction in which the mass m. would have moved during the next instant had it not been subjected to this arbitrary displacement, and let the distance in each case be pre. cisely equal to the distance which the body would have moved dur- ing the next instant had it not been subjected to displacement. Then by the theorem in virtual velocities that ] FSp &t = change in the living force, we shall have for the masses m l m k , = OJL tor m, mmmm & in, JL / h 1 adding we get (a) 1 1 5 6 These equations involve the masses because F^ are forces on unit mass. Now it is known that the change in the living force of a system is equal to the work done on the system and since work equals force X distance, we shall get for the change in the living force V / f * i sl-i% i 1 \AJ\J Equating these two values of S T, we get, = which is Lagrange's Generalized Equation. If now we suppose the forces to be resolved along the three coordinate axes the above equation can be easily made to assume the form, where X, Y, Z are the total components of the forces along the coordinate axes. Let us assume a certain function U (Potential Function) which is independent of the time t, such that W SU dU -> ^ = -^JL , ~~^i = JL , ~ == & , dx dy dz then by substitution equation (2) becomes ifc ^ + ' - etc -) = 2 ( m w Bx + '" etc *)' Now the left hand member of this equation is the total varia- tion of 7, or 8V. Since T (Living Force) = \ mi?, &T= mvSv, but 1 d 2 x dv m W^^ m dt and adding ^ dv m -j-z ox = m -j- ox + mvov o T at at now mv ^i for Hence (Sx) = mvS I -5 \ mv8v. or (3) SU=j t Let us suppose I 7 to be a function of the independent variables q l9 q 2 , - - -, etc., then the variation of T is etc., 3s &t -3j*9i+ These values substituted in (3) give the equation and since the ^'s are independent we can equate the like variations and obtain the following partial differential equations : d J?iL( m ^\__^ dq l ~~ dt \ dq l ) dq l etc. etc. etc. which become dU d / dT\ dT (4) dq l ~ dt\ dq( ) dq l etc. etc. etc. Since ds -^dsdq ds dv ds v = / - -f- and -, == . dt ^^ do dt dt dq, dq, But J mv 2 = T, therefore dT dv ds , = mv =-7 = mv = These equations are known as Lagrange's Canonical Forms, and in deriving them we have assumed that all points of the system have been expressed in terms of , and k independent variables q l - q k . Since there are &n coordinates altogether in the system, (ic x y l z l9 -, x n y n z n ) this assumes that there are (%n k) equa- tions of condition. II. CANONICAL FOKMS OF HAMILTON. Let us still regard T as expressed in terms of q , , q h , %i > '"> Qk) anc * wrl * e _dT _dT T was originally a homogeneous function in regard to dx l dx 2 9 and since dx , dy , dz , are connected with ^J , q' 2 , by linear equations, T regarded as a function of q and ^' is homogeneous and of the second degree in q[ , ^ It, therefore, satisfies Euler's equation, or dT dT Taking the variation of T 2S2' to ,, ) = and by direct variation a? subtracting but Equating like variations we get dT H etc. _ dq ~ dq etc. where , 2 Now let ZT= jf C/", where ZT== constant independent of then dU and equations (4) give subtracting Again 10 dU d Pl dq l ~ dt dt = dH Now U is supposed not to contain q[ , p l , or t , hence d Ujdp l = , and we have just shown in equations (4) that dT dt hence by substitution We thus have the systems of equations (5) dt dH dH dt d qi , dH dt dH dt These equations reduce the system of k differential equations of the second order to 2k differential equations of the first order. If we call p t and q. conjugate independent variables, Hamilton's reduction may be stated thus : " Hamilton's Canonical Forms arise from finding two series of variables in terms of which the coordinates x , y , z , can be expressed. The total differential of any one variable with regard to the time is equal numerically to the partial derivative of a certain determinate function, H, with regard to the conjugate variable." 11 in. METHOD OF JACOBI AND ITS APPLICATION TO Two BODIES. CANONICAL CONSTANTS. Let us suppose these 2k equations of Hamilton to be integrated, then we will get 2 Ic constants of integration c x c a , and let us take the partial of ZTwith respect to c, , since H will be a function of the c's, whence _ dp ~~ ~ and by substituting from (5) this becomes _ , \ "' i "' dc i 1 ' / , dp, \ dp, dq, t'dc l d ( da, = ^A^ But 2T= Z^i(^i)/ d dq l d f dq, \ ~M^ + ''7* Now ZT= ^ T C7", and substituting, then, ) dfdq ~ P H Integrating with respect to , we get Assume and multiply by cZc x , then and 8$-- 12 Taking the variations of S directly we get and since the ^'s are independent (6) We have ~di = =~dt+ etc. -.Pio etc. dS dS ' and We here consider If as a function of the 2& constants ^ , but independent of t. The equation (Jacobi's) dS d$ when integrated will give S containing the k constants q l - q k , and since the partials of S with respect to these k constants are to be put equal to k constants by (6), we shall have introduced upon integrating this last series of k partial differential equations of the first order, 2k constants altogether. To integrate such a differential equation of the first order, we have need of Euler's Transformation, which is derived as follows : Suppose Z = (ojjO^cCg ) and we desire to integrate 0. 13 Assume y = z x^ dzjdx^ = z x^ , then dy = dz x 1 dx[ x( dx 1 , but from the equations in , we have dz dz . and by substitution dx~ dXl + ' (d* i \ = I -j- dx~ + ) x. ax, . \dx 2 ^ ) Therefore dy dy dz dx(= ~ x ^ ^ 2 = ^ 2 ) '" >etC ' ) and ^ = = dx l dx l ' These values substituted in the original differential equation, gives Our new equation contains the same number of variables as the original equation, but the variable x l is replaced by ( dyjdx^) . If this latter is a constant by the conditions of the problem we have thus removed a variable. It is easily shown that the equations of undistributed motion for two bodies are, in which problem 14 d*z_ "3? " r TT /77 7"7" JJ_ sss _/ (_/ :^ > r Now Jacobi's equation is and hence ^T must be expressed in terms of the new variables. Let dx ft-* .-s ?2 = y, then jJ_J, and T = |m [(,;) + ( ? ;) 2 + &)*] 6? ?a = ?,- S by substitution or i Now transform to polar coordinates by means of the equations x = r cos a- cos z> y = r cos ^ 2 . da ) J" r ~' 15 Now apply Euler's transformation by letting S' = S -f cut , then dSJdt = a, [a is a constant of integration] and substituting Solving we get again, let /S"' = S" a^, then dS'jdv=a l and our equation becomes which can be written Put the left member = a\ , then r 2 or - and Hence the complete integral gives X r \ 2& 2 a 2 C v \ o> -J2a+-- dr+ -Ja 2 2 L \ r r" i \ COS ^o Now ^ = S' - a^, and ^ = S + at, r I 2F of .S-^ir + J^ ^2^ + -^ r a 2 2 - J \ 2 cos 16 Put then a# as i and dr = + ^ - /^=O, .-.^=11. Let ^.P (7 be a right spherical A in which / PA Ci^ AP r) and P C = & , then sin where U = - . dt 2 dx r When a third body is added to the system U is of the form where It is the Perturbing Function. The question, therefore, is 18 to find what change must be made in the canonical constants in order to replace H by ( H If) in the equation just solved. From Hamilton's Forms, eqts. (5), we have dq l _dH ~dt ~ dj^ ' and if ff= H R, becomes dq l _ d(ff-R) _ dH _ dR dt ~ d Pl " d Pl ~ d^ (9) likewise ^i= - dj ^= d(H-R) = _dH dR dt ' d qi ' d qi Wi + d%i' Considering p l and q^ as functions of the constants and , we regard the constants as variables and find what variations must take place in them so that H may be replaced by (H R), that is, p^ and q, must satisfy (9) . Assume *, ^ --- ft, then i 4- ^ ^ dt da ' dt d/3 dt ^ ^a a^ ^\ ^ '^ ^ ' dt ) now equation (5) gives d ^- dH aid *' = == -^ - d/llU _ a^ ^Pj dt which substituted in (9) gives 'a dt Sft dt (10) dR ^ ( dpi da dp l dfi a dt d/3 dt 19 Since we can find p and q exclusively in terms of the a's and /3'Sj and also a and fi in terms of the />'s and q's , we can apply Jacobi's Theorem which states that, (a) -~- = -~ (c) ~ -~^ ^ a k ^9.i ^fik ^Pi whence do, d/3 da, da dp. d 8 -^r L== jr-'> -^=5' -^ = ^r- and da dp l op dp l da dq l and making these substitutions in (10) dj3 da da d/3 d pl ~ *- d H f d/3 da da da df* di ' If we express R in terms of the p's and ^'s then by Calculus of Variation, we have and substituting from above the values of dR /dp and dltjdq, we get da da d3 d * da da which can be written 3/3 80 da I8a da dfi Since a and /3 can be expressed in terms of the p's and #'s , let us assume then and by substitution in &??, we have If now J? is expressed in terms of the a's and yS's , then and equating like variations, we have dt etc. dt etc. These are Jacobi's equations and they give the total variation of the constants in terms of the partial of the Perturbing Function, when the latter has been expressed as a function of the constants (a, /3) and the time (t). V. DIFFERENTIATION OF THE EQUATIONS CONTAINING THE CANONICAL CONSTANTS. Solving equations (8) we find (12) cos i = - 1 , and let e = TT nr , P , then e = /^ 4- /3 2 since n = = . By differentiation of the first of these equations 21 da 1& da. 2a 2 da da da dt (13) ? J a/3 ' in like manner we can get de a i/l e 2 sin ij-. = - ,- /:== cos i ^~ =-3- dt Jc i/a i/l e 2 L ^2 3ft ^n ^ W de dt kdR a* da dR Since e is the only equation in (12) containing y8, and 7? being a function of the a's and /3's , then Likewise but dR _dR de _k_ dR ~d$ = ~~~de~'d@ =: tf' ~de~' de dR dTr dR dfl 3j3 l dir dfi l dfl ^(B l ^T _ dfl _- [ ' de a de ' dR dR dR dR also dR dR de dR dir de ^/o~ == ~^7~ ' ~^Q I 5Z7 ' xa and ~ o = 1 , Sir 1. 'W dR dR dR = ^r + de r dTr' 22 In a similar way we get the following as the partials of R with respect to a , a^ , and a 2 respectively : a (l-e 2 ) dfi__3a 1 dR 1 i/l e 2 dR 1 cos i da z~ &!/ G de kVa i/l e 2 sin i & ' Put n = ^/a^, and substitute these values in (13), then (14) da_^ di ~na'~de' dt no 2 i/l e 2 sin i ^ d'jr 2 ^ no 2 i/l e 2 ^e ^ + (Zi 5^ wa 2 a?r -1 -yT= l-l/l-e 2 a^ e" sin + a -2 dt ~~ na + \/i- 1-i/l-e 2 de 23 VI. TRANSFORMATION OF EQUATIONS EXPRESSING THE PERTURBA- TIONS AND THE VALUES OF THE VARIATIONS. The perturbations can be expressed in another form by the fol- lowing substitutions: Let the perturbing force which m exerts upon m at any instant be resolved into three rectangular components as follows : (1) (w'/l + m) R' is the component along the radius vector of m 1 reckoned positive away from the sun. (2) (m'/l + fn>) &' is the component perpendicular to the radius vector and in the plane of the orbit, positive in the direction of motion. (3) (m'/l +S" ["cos u sin * 1 + W \ cos i 1 . m dz By means of the above expressions we can express the partials of the Perturbing Function ( R ) contained in equations (14) in terms of the components R' ', $', and W. As an example we shall find the value of dRjda: Now - [ 1 \ d ~y~ ~~fa[_dr 'da] + "^ \_~dr 'da da 25 From the properties of an ellipse, r = a(l e cos _Z?), dr r . k ... . . = , also n = -; = mean daily motion , da a a* then jE' e sin E nt -f e TT , v II + e E tan 2 = >lr^ tan ^' and (16) cc = r {[cosll cos w] [sinfl sinit cos*]}, y = r { [ sin.fl cos w ] [ cos U sin w cos i ] } , z = r {[sin w sin i]}, whence we express x, y, and g in terms of the constants a, a 1? a 2 , and /3, ^, /3 2 , and ; likewise, dx/dt, dy/dt^smd dz/dt. By differentiating (16) we have - dr = fcos H cos u sin H sin u cos 1 = cos R' X, y -*?-= [sin fl cos ^ + cos fl sin u cos i~] = cos jff' J", ^- = [sinwsin*], =cos^'Z, and substituting in (15) Now we substitute in this equations the value of d Rjdx , d Rjdy and dRjdz derived in the beginning of this article; and it re- duces to dE r ( m' \ da ~~ a \l +m )' 26 for the terms containing the squares of the cosines and their prod- ucts can be reduced by the formulae cos 2 a -f- cos 2 /3 + cos 2 7 = 1, cos 2 a -f cos 2 /3' -f cos 2 7' = 1 , cos a cos a! + cos cos /3' + cos 7 cos 7'= , where the cosines are the direction cosines of two lines. To find dRjdi we have r da 2m' na 3 dt de m' na 2 i/l e 2 -ji = ^ ^ [ sin v2i dt \ + m 1 + (cos v + cos .#) S f ] di m na dt = i^'yT=7 2[rGOSuW l mi -dt = - = ^- -- Ap--[-cos^' dt 1 -f m 1 4- #'(1 +r)/psinv] or c?e m' _ _ . -j-=-2an^ -rJB f + 2sm 2 J-^ +2 sm 2 ^ , , dt 1 + m 2 eft 2 dt where e = sin . VII. DR. G. W. HILL'S * FIRST MODIFICATION OF GAUSS'S METHOD. If the orbits do not intersect each of these differential coefficients may also be obtained in the form of an infinite series arising from the expansion of the Perturbing Function to terms of the first order with respect to the disturbing forces. Since the series contains only terms of the form A*(iM + i'M') in which A is a con- stant and i and i' positive integers, it follows that the secular l On Gauss's Method of Computing Secular Perturbations, by G. W. Hill, Astronomical Paper of American Ephemeris, vol. I. 28 portion of any differential coefficient will be that corresponding to i = and i' = . If we consider, for example, the coefficient de/dt we will have and the part independent of Jf' will be the part of this series independent of M is hence and this is the secular part of the perturbation. The computation of the secular part of the perturbations is thus reduced to evaluating the double integrals, 2lT ~dMdM'. at etc., from the expressions found for them from equations (17) of the last article. The integration with respect of M' can be effected rigorously in terms of elliptic integrals of the first and second species, but that with regard to M can only be approximated to by mechanical quadrature. This quadrature is more accurate if made with regard to E, and we hence transform to this variable by the usual formulae. The variable M' is replaced by E' also for purposes of symmetry, when we shall have In which we have written for brevity _1_ C 2 " ar etc. 29 Gauss's method of effecting the integrations ( b ) consists in re- placing the variable E' by a new variable T which is connected with E' and ten new auxiliaries by the following equations : ^sin E' = ft -f ff sin T + 0" cos T, jVcos E' = a + a' sin T -f a" cos T, If = 7 -f ry' S in I 7 + 7" COS I 7 . The values a , /3 , 7 are so taken that the coefficients of sin T and cos T vanish in the expression A 2 (7 + 7' sin T -f 7" cos T) 2 [in which A is the distance between the two bodies], which hence takes the form This substitution thus finally reduces the integrals of (b) to the form a sin 2 T -f b cos 2 T r* Jo dT, in which and 5 are independent of T but involve G, G', and 6r". This integral can readily be broken up into elliptic integrals of the first and second kind of which the modulus 2 G' + G" = G + G" In the memoir by Dr. G. W. Hill the steps of this reduction will be found given in detail and also very exact tables for effect- ing the computation. These quantities of the tables are the func- tions of the elliptic integrals met with in evaluating ( c ). They are tabulated to the argument ( = sin -1 c ) , and are published to eight decimals, having been computed to ten. When the values of JS , S Q and W Q have been found, a direct quadrature of ( a ) will be resorted to to effect the second integra- tion. It is probable that no accuracy is lost by our inability to exactly integrate these expressions since, as is well known, the order of error committed cannot in any of the coefficients exceed a 30 power of the eccentricities and mutual inclinations of the orbits one has, than the number of parts into which we divide the orbit of the disturbed body. VIII. COMPUTATION. The following is an application of Dr. Hill's method to the action of Jupiter upon Mars, and the elements employed are from Dr. Hill's "New Theory of Jupiter and Saturn," pp. 192, 558. Mars. TT = 333 17' 51".74 i = 1 51 2 .24 H = 48 23 54 .59 e=0.09326803 n = 689050".784 log a = 0.1828971 1 m = Jupiter. TT' = 11 54' 31".67 ' it = 1 18 42 .10 fl'=98 56 19 .79 e = 0.04825511 ri = 109256".626 log a = 0.7162374 1 m = 3093500.0 1047.879 [Epoch = 1850.0 G.M.T.] From these elements the preliminary constants become /= 126' n = 149 47 n' = 188 22 ^T = 321 24 J5r=321 24 6".38 4 .37 45 .43 28 .27 9.62 log k = 9.9999971 log # = 9.9998667 log c = 8.7995614 c = 0.06303204 If the orbit of Mars be divided into twelve parts with regard to the eccentric anomaly, the values of the auxiliary functions cor- responding to the several points of division will be as given in the following tables. A rough test of these values is found by com- paring the sums of the functions corresponding respectively to the odd and even points of division of the orbit : these are given at the foot of the columns. Sums marked thus (*) indicate that the corresponding numbers have been added instead of their logarithms as given by the points of division. 31 A test of the perturbations in the plane of the orbit is afforded by the condition that, since raa-} L & Joo , sin cos = , this residual is found to be + 0.000,000,000,003,2. The computation has not been duplicated, but various checks on the accuracy of the work have been employed as the computation progressed. E Logr V A log B 6 0.1403760 o o' o.'bo 29.5201397 0.9162454 327 64L59 30 0.1463201 324724.62 29.7305725 0.9338723 355 454.78 60 0.1621568 644446.64 29.8340289 0.9418636 22 20 51.41 90 0.1828971 9521 5.91 29.8101781 0.9397444 49 30 45.26 120 0.2026919 124 31 47.16 29.6691083 0.9283109 77 529.50 150 0.2166314 152 34 23.40 29.4449238 0.9092471 105 33 19.44 180 0.2216237 180 0.00 29.1903005 0.8855869 135 21 49.56 210 0.2166314 207 25 36.60 28.9697704 0.8627418 166 51 33.40 240 0.2026919 235 28 12.84 28.8461195 0.8485520 199 55 33.96 270 0.1828971 264 38 54.09 28.8598713 0.8498206 233 38 36.66 300 0.1621568 295 15 13.36 29.0110391 0.8666375 266 35 28.90 330 0.1463201 327 12 35.38 29.2554191 0.8917695 297 46 46.75 8 1.0916971 900 0.00 176.0707360 5.3871961 1028 25 54.92 S' 1.0916972 1080 0.00 176.0707353 5.3871956 1208 25 56.29 E Logfir A | G \ G' 0.1016599 27.008366 2.448742 27.0064625 2.4695932 30 8.5335913 27.006191 2.661350 27.0061438 2.6618714 60 9.8433695 27.007668 2.763330 27.0066000 2.7737057 90 0.4413039 27.011682 2.735464 27.0074721 2.7765140 120 0.6339503 27.014415 2.591662 27.0078882 2.6581506 150 0.5856433 27.013255 2.368638 27.0074658 2.4330414 180 0.2641553 27.009344 2.117924 27.0066063" 2.1522696 210 9.2384099 27.006392 1.900346 27.0061375 1.9039682 240 9.5616845 27.006977 1.776111 27.0064375 1.7842143 270 0.3111659 27.010167 1.786673 27.0071677 1.8310715 300 0.5312995 27.012575 1.935433 27.0075528 2.0032709 330 0.4767380 27.011664 2.180724 27.0071925 2.2348548 S 11.8660187* 162.059344 13.633201 162.0415473 13.8412043 S' 11.8660198* 162.059350 13.633194 162.0415794 13.8413213 32 E G" 6 Log-R Log I' Logfl o / // 0.0189480 17 39 53.64 0.03165480 0.31498607 0.22325501 30 0.0004748 18 17 56.54 0.03402429 0.31811121 0.22675996 60 0.009308F 18 43 15.10 0.03565127 0.32025563 0.22916456 90 0.0368400 184858.26 0.03602459 0.32074753 0.22971607 120 0.0599625 18 28 29.55 0.03469754 0.31899872 0.22775521 150 0.0586154 17 39 45.34 0.03164635 0.31497491 0.22324250 180 0.0316073 16 30 39.87 0.02757623 0.30960115 0.21721374 210 0.0033673 15 24 39.12 0.02396320 0.30482483 0.21185327 240 0.0075643 145527.27 0.02245002 0.30282277 0.20960581 270 0.0413999 15 15 16.60 0.02347167 0.30417461 0.21112339 300 0.0628166 16 214.88 0.02598814 0.30750244 0.21485859 330 0.0496602 165332.60 0.02889110 0.31133795 0.21916250 8 0.1902075 10220 0.31 0.17801800 1.87416678 1.32185292 S f 0.1903576 102 20 8.46 0.17802120 1.87417104 1.32185769 E LogN LogP LogQ Log V Ji 8.3476455 5.799087ft 7.1391282 7.1387519 27.0207829 30 8.3623563 5.8175271 7.1576461 7.1576367 27.0065656 60 8.3954328 5.8524494 7.1929779 7.1927933 27.0099909 90 8.436602T 5.8931978 7.2342423 7.233512T 27.0290204 120 8.4742977 5.9283888 7.2695991 7.2684105 27.0492441 150 8.4991682 5.9492923 7.2899853 7.2888220 27.0526533 180 8.5057537 5.9513989 7.2909894 7.2903609 27.0334422 210 8.4928482 5.9346398 7.2731849 7.2731178 27.0094522 240 8.4633467 5.9029905 7.2413630 7.2412123 27.0106059 270 8.4239466 5.8638326 7.2029257 7.2021014 27.0363166 300 8.3844575 5.8269715 7.1668219 7.1655727 27.0521291 330 8.3560126 5.8027962 7.1428979 7.1419112 27.0423359 8 50.570933S 35.2612859 43.3008794 43.2971015 162.1761951 S / 50.570933$ 35.2612856 43.3008821 43.2971011 162.1763440 33 E J, J* LogJF 3 Log^ 3 -Bo 0.20899495 , +0.30968486 0.7664249 9.059339671 0.011523703 30 0.03630764 0.02674098 9.9823906 8.3442271n 0.011964426 60 +0.15284448 0.36413193 0.637279771 8.9312244 0.012921849 90 +0.31348410 0.61207986 0.936246971 8.9588266 0.014185451 120 +0.40064234 0.70414532 1.032570171 8.3275956n 0.015433305 150 +0.38540725 0.61566127 1.008416671 9.169522071 0.016299428 180 +0.26823871 0.37034154 0.8476726n 9.2218350n 0.016504151 210 +0.08088125 0.03392182 0. 3347999n 8.766871771 0.015978577 240 0.12278036 +0.30345688 0.4964372 8.8714541 0.014896494 270 0.28345025 +0.55139873 0.8711779 9.0346961 0.013593041 300 0.35632065 +0.64347029 0.9812447 8.2963250 0.012429948 330 0.32638729 +0.55499846 0.9539640 8.9732022n 0.011688529 8 +0.13362957 0.18200676 3.6043738* 0.123045628* 0.083709450 8' +0.13362742 0.18200674 3.6043746* 0.123045667* 0.083709452 E S W Sn Rn +0.00008004999 +0.00041903914 5.7629857 30 +0.00001088807 0.00003989400 4.8906308 7.6305418 60 0.00007058128 0.00056153595 5.686533971 7.8866984 90 0.00013852398 0.00104078831 5.958627971 7.9689460 120 0.00017072125 0.00130819274 6.029595971 7.9232977 150 0.00015776366 0.00121033849 5.981375271 7.6945110 180 0.00010615130 0.00073761030 5.804301671 210 0.00003427562 0.00006864982 5.3183538W 7.685876771 240 +0.00003689362 +0.00053477066 5.3642593 7.907922871 270 +0.00009184620 +0.00088606326 5.7801639 7.950419571 300 +0.00012131643 +0.00094343743 5.9217628 7.8698431n 330 +0.00011863616 +0.00076351967 5.9278969 7.620409871 S 0.00010919379 0.00071009176 0.000054778522* +0.0005847741* ' 0.00010919284 0.00071008769 0.000054774361* +0.0005847090* 34 E R sin v + S (cos v + cos E ) -*.*, +fS .*n, Log W sin u. Log W cos u. \ a cos 2 / +0.0001601000 0.0115237021 6.607402471 6.0323895 30 +0.0064980855 0.0100466448 5.4290192 5.4698495w 60 +0.0116214735 0.0056380423 6.0040176 6.742246771 90 +0.0141365355 +0.0010459980 6.566602971 6.989651471 120 +0.0128965787 +0.0084589862 6.997256971 6.9298457n 150 +0.0077844290 +0.0143154744 7.0724419n 6.419187071 180 +0.0002123026 +0.0165041521 6.8529748w 6.2779619 210 0.0072998748 +0.0142155824 5.705474971 5.6648794 240 0.0123115656 +0.0083813594 6.2543292 6.702177471 270 0.0135423545 +0.0010841018 6.1672230n 6.9414069w 300 0.0111295720 0.0055182011 6.784158371 6.857991871 330 0.0061276125 0.0099498995 6.861294271 6.3702507n S +0.0014493172 +0.0106645521 0.00243928405* 0.00233062987* sr +0.0014492082 +0.0106646123 0.00243923736* 0.00233061181* E --KO If ra' is left indefinite, the resulting values of the dif- 0.020897817 ferential coefficients are : Loe coeff. 30 0.021996061 [^1 = + 165"71042m' 2.2193498 60 0.024638500 L at Joo 90 0.028370902 [dx~\ mf 4.1164137 120 0.032306045 dt Joo 150 0.035231951 180 0.036086917 g] M = 268.8244 1m' 2.4294687W 210 0.034538413 240 0.031182353 f^l = 8712.3580 L^Joo ro' 3.940135771 270 0.027186081 300 0.023700582 ['s] w =+ 13069 - 611 m' 4.1162627 330 -0.021488837 S 0.168812214 L<^Joo == ~ 19834 ' 253 m f 4.286327471 S' 0.168812245 35 If the above value of m be employed, we get de 00 00 0.15813891, 12.476782, =- 0.25654148, = - 83142788, oo 00 + 12.472445, =-18.450846. oo The values of Newcomb are stated on page 378 of his " Secular Variations of the Orbits of the Four Inner Planets." Those of Le Verrier are found in the Annales de 1'Observatoire de Paris, Tome II., page 101, and Tome VI., page 189. The results of Le Verrier have been reduced to the above value of m', the three sets of values then compare as follows : [ Results of Kesults of Method of Le Verrier. Newcomb. Gauss. ^ I =+ 0.15810 +0.15818 + 0.15814, ~ ^ J e\^r-\ =+ 1.16323 +1.16372 + 1.16328, oo ~ =- 0.25640 -0.25655 - 0.25654, dt Joo - 0.26873 -0.26850 - 0.26850, -18.45085. 36 BIOGRAPHICAL. The writer of this thesis was born in Baltimore, Md., August 11, 1872. Prepared at Friends Elementary and High School, Baltimore, Md., he entered the Johns Hopkins University in 1889, from which institution he received the A.B. degree in 1892. Graduate student in Physics, 1892 to February, 1893, when he accepted a position in the U. S. Coast and Geodetic Survey ; 1894 1895, Principal of Martin Academy, Kennett Square, Pa. ; 1895, to date (including absence on leave 1899-1900), Professor of Mathematics in Temple College, Philadelphia, Pa. While teach- ing he has pursued graduate work in Astronomy, Mathematics, and Physics at the University of Pennsylvania. Also pursued graduate work in Physics and Mathematics at the Johns Hopkins University, 1899, to February, 1900. 594329 UNIVERSITY OF CALIFORNIA LIBRARY