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 = <f> (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 <r sin v 
 z = r sin cr whence 
 
 2 1 /dxSV 1 /#V1 >^ 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 
 
 
 = + ^ - 
 
 /^<r <&T 
 
 V. 7T3 
 
 cos^o- 
 
 cos cr 
 
 where r x is the least root of the equation 
 
 Let r = r 1 , then t = /3 , or /3 = Time of Perihelion Passage, 
 
 2 
 
 If PJ = smallest root 
 
 largest " 
 
 then 
 
 and 
 
 r 1 _a(l e), ^^ 
 
 hence equating we get 
 
 2a= - , 
 a. 
 
 - e . 
 
 From the value of /^ we see that a 2 /cos 2 a = a 2 determines the 
 maximum value of cr (i. e., <7 = i) whence 
 
17 
 
 1 = a 2 cos cr = k i/5 i/l e 2 cos i , 
 and when 
 
 cr=0, i>=O, .-.^=11. 
 
 Let ^.P (7 be a right spherical A in which / PA Ci^ AP r) 
 and P C = & , then sin <r = sin i sin 77 , and cos <rd(r = sin i cos 77^77 
 which substituted gives 
 
 S*<r d<T S*cr C?<7 
 
 ** I ~n ~^r '** / ("TTT cos 2 f 
 
 / \ <*l -- 2~ 1 \ a l 1 -- 2 
 
 J o \ 2 cos 2 cr / o \ 2 \ cos 2 cr 
 
 COS crjcr 
 
 / o 
 V cosh 
 
 er 
 
 rjcr Ft 
 
 ^=f^ = 
 Sln 2 ^ J 
 
 If the body is at perihelion, then 77 = co , and y5 2 = co , or 
 5 2 = TT il . Therefore our six constants of integration or 
 canonical constants have the values 
 
 (8) 
 
 * i/a 1/1 e 2 cos * 
 
 IV. 
 
 VARIATION OF THE CANONICAL CONSTANTS 
 
 AND JACOBI'S EQUATION. 
 The equations of motion for two bodies are of the form 
 
 d 2 x dU u 
 
 -Ta = ~^ > 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 +<m)W is the component perpendicular to the 
 plane of the orbit, positive northward. 
 
 Hence each of the variations of R along the coordinate axes will 
 be made up of three parts, and will be determined by the equa- 
 tions, 
 
 m dR cos + cog 
 
 m! dx 
 
 ~ ^ = R' cos R'Y + & cos S'Y + W cos TFT", 
 
 m' % 
 
 .#' cos .#' + #' cos tf'Z + IF' cos TF'Z. 
 
 m' dz 
 
 The values of the cosines can be obtained from the following 
 spherical triangles : 
 
 Let the plane of JTJTbe the ecliptic (the X axis passing through 
 the vernal equinox) and T the point where it is cut by the plane of 
 the orbit of m; OR ', the radius vector of m, then the 
 
 gives 
 
 cos R' X cos U cos u -f sin U sin u cos (180 i) , 
 
 = cos ft cos w sin ft sin u cos ^ . 
 
24 
 
 Let /S" be drawn in the plane of the orbit perpendicular to 
 OR, then / #OT = 90 + u, and AJTT#' gives 
 
 cos S'X= cos fl cos (90+w) + sin O sin ( 90+w) cos (180 - *) , 
 = cos H sin u sin fl cos u cos i . 
 
 Let W be drawn perpendicular to the plane of the orbit, 
 then / TF'TX= (90 - ), and AXT TF' gives 
 
 cos TF'X = cos n cos 90 + sin fl sin 90 cos (90 i), 
 = + sin U sin i . 
 
 On substituting these cosines in our first equation above, we 
 have 
 
 - ? -- 5 = It' fcos u cos U sin u sin fl cos 1 
 m dx 
 
 4- /S" [ sin w cos O cos u sin H cos *] 4- W [sin U sin *] . 
 In like manner we can derive the equations 
 
 - -, -- ^ = H' fcos u sin U 4- sin w cos U cos i 1 
 m dy 
 
 4- /S" [ sin w sin ft + cos w cos U cos ] 4- W [ cos U sin i~\. 
 and 
 
 - ^ -= = j?T sin w sin i 1 4- >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 
 
 <te~| , f^ ^1 , f 
 
 di \ + [ d y '& J L 
 
 - 
 
 dz di 
 
 And from (16) we get 
 
 - = r [sin u sin O sin z 1 
 ^1 
 
 G y 
 
 -~ = / [ sin M cos U sin ^] 
 
 dz 
 
 - = r [sin w cos 1 1 
 di 
 
 and knowing dfi/dx, dfi/dy, and dRjdz^ dRjdi reduces to 
 dE m 
 
 di ~ 1 -f m 
 
 r W sin u . 
 
 The other partial differential coefficients can be found in a 
 similar manner. If these coefficients be introduced in equations 
 (14), we obtain the following variations of the elements : 
 
27 
 
 (17) 
 
 j^._^r sin ^ + ^i 
 
 1 + m V T^7 2 L > 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 <f> . 
 
 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 <f> / 
 
 
 
 
 
 +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