LIBRARY 
 
 OF THE 
 
 UNIVERSITY OF CALIFORNIA. 
 
 RECEIVED BY EXCHANGE 
 
 Class 
 
SECULAR PERTURBATIONS 
 
 ARISING FROM THE 
 
 ACTION OF SATURN UPON MARS 
 
 AN APPLICATION OF THE METHOD OF ARNDT 
 
 A THESIS 
 
 PRESENTED TO THE FACULTY OF PHILOSOPHY OF THE 
 UNIVERSITY OF PENNSYLVANIA 
 
 BY 
 SAMUEL GOODWIN BARTON 
 
 IN PARTIAL FULFILMENT OF THE EEQUIREMENTS FOR THE 
 DEGREE OF DOCTOR OF PHILOSOPHY 
 
 PHILADELPHIA 
 
 1906 
 
SECULAR PERTURBATIONS 
 
 ARISING FROM THE 
 
 ACTION OF SATURN UPON MARS 
 
 AN APPLICATION OF THE METHOD OF ARNDT 
 
 A THESIS 
 
 PRESENTED TO THE FACULTY OF PHILOSOPHY OF THE 
 UNIVERSITY OF PENNSYLVANIA 
 
 BY 
 SAMUEL GOODWIN BARTON 
 
 IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE 
 DEGREE OF DOCTOR OF PHILOSOPHY 
 
 PHILADELPHIA 
 1906 
 
PRESS OF 
 
 THE NEW ERA PRINTING COMPANY 
 LANCASTER, PA. 
 
INTRODUCTION. 
 
 The paper upon which this thesis is based is entitled " Re- 
 cherches sur le calcul des forces perturbatrices dans la theorie 
 des perturbations seculaires" by Dr. Louis Arndt. The paper 
 is published as a bulletin of the " Societe des Sciences Naturelles 
 de Neuchatel," an extract from tome XXIV, 1896. 
 
 Aside from the application of the author, as a check upon his 
 formulae, mentioned in the paper, so far as is known, no applica- 
 tion of this method has been made. The purpose of the author of 
 this thesis is to apply this method of computation and to compare 
 it with that of G. W. Hill and thus see whether Arndt's method 
 has the advantages claimed for it by its author. 
 
 I. 
 EXPRESSIONS FOR THE SECULAR PERTURBATIONS. 
 
 Let R, U, Z (each multiplied by the factor containing the masses, 
 i. e., m I(L -\- m) be the components of the perturbing force; 
 R acting in the plane of the orbit of the perturbing body and in 
 the direction of its radius vector, r, positive if r is increased ; U in 
 the plane of the orbit perpendicular to the radius vector, positive 
 in the direction of movement ; Z perpendicular to the plane of the 
 orbit, positive northward. Let fl be the longitude of node, i the 
 inclination of the orbit to a fixed ecliptic, a the semi-major axis, 
 e the excentricity, /A the mean daily motion, a> the angular distance 
 of perihelion from the ascending node, v 9 E, M, the true, eccentric 
 and mean anomalies and u = CD -j- v. 
 
 In accordance with the usual method of varying parameters we 
 have the following equations for the variations of the elements. 
 
 3 
 
 165260 
 
4 
 
 map sec<t> _ . 
 
 sin (8fl) = j^ - - f sin u Z, (S) = -, r cos 
 
 [sin B + (cos v + cos 
 
 (So)) = (8a>,) COS i(Sfl), 
 
 (I) 
 
 * + 2 sin' (&0 + 2 sin 
 
 The secular part of these expressions will comprise those terms 
 which are independent of the positions of the bodies in their orbits, 
 that is of their mean anomalies M and M ' . Each of the above 
 variations is seen to be a function of 7?, 7, and Z which in turn 
 are functions of M and M' . They may therefore be expressed in 
 a Fourier's series, i. e., in a series of sines and cosines of multiples 
 of M and M ' . Now the terms of this expansion which are inde- 
 pendent of M and M' give the secular variation. By the known 
 properties of the series the only constant term with respect to M 
 is given by the integral 
 
 and the part of this which is constant with respect to M' will be 
 
 Thus the secular part of any variation (Se) for example is 
 
 TO '" V C S * i r r [sin vR + (cos v + cos E) U^lMdM '. 
 
 1 -|- m 47T JQ JQ 
 
 Since the limits of integration are constant, sin v, cos v and 
 cos E, may be treated as constants in the integration with respect 
 to M' i and this integral may be written 
 
 m'a*p cos <f> 1 ** . \ f- 
 
 - ^ fsm v ~~ RdM 
 
 1 + m 27rJ L 27rJ 
 
 -f (cos v -f cos E) ~ I**" UdM']dM. 
 
 " rjr Jo 
 
Treating the other variations in the same manner we see that 
 the integration with respect to M' requires the three integrals 
 
 1 / 
 
 fcf 
 
 The integration with respect to Jlf can be obtained only by a 
 direct quadrature. Since this can be obtained more accurately in 
 terms of E we transform M and M' in our expressions into E 
 and E '. Then since 
 
 M=E-e sin E, dM= (1-6 cos E)dE=-dE, 
 dM' = (l-e r co%E')dE'. 
 
 Our desired integrals which which we shall now designate 
 7? , Z7 , Z respectively are 
 
 ~\rR(\ - c' cos .E")dtf', ^ Par 7(1 - e' cos 
 (2) 
 
 -^ f ^ r 2 ^! - e' cos j0')cLtf' 
 
 where the factors a and r have been taken with the integrals. 
 
 II. 
 PRELIMINARY CONSTANTS. 
 
 Let the orbit of the perturbing body m be referred to that of 
 the perturbed body m. Let the distances of the perihelia from 
 the intersection of the orbits be II and II' and their mutual incli- 
 nation J. Let K and K' be the distances of the intersection 
 from the nodes. Solving the triangle thus formed we obtain 
 IT, K' and J in terms of i, i' and (ft' ft) whence we get II and 
 IT from the relations 
 
 n = TT ft - K, IT = if ft' - K'. 
 
 Let L' and B' be the longitude and latitude of m referred to 
 the orbit of m and the intersection. Let 77, % be rectangular 
 axes, origin at the sun, 77 and f lying in the plane of the orbit of 
 
6 
 
 w, f passing through the body m. Let I be the longitude of m. 
 The coordinates of m are then given by the equations 
 
 I H -f v, f = / cos # cos (L r - Q, 
 
 T/ = r' cos .7?' sin (' Z), " = / sin B. 
 
 Now from the right triangle formed by the intersection, m\ and 
 the foot of the perpendicular from m to the orbit of m we have 
 
 cos L' cos E' cos (II' -f v), sin U cos .Z?' = sin (II' + v') cos J] 
 
 sin 5' = sin (IT + v) sin 7. 
 To simplify the formulae, write 
 
 A sin A' = sin II' cos <7, vl c = Ad cos (J/ + II + v), 
 
 4, = Ba! cos <' sin (^' + II + v), 
 J. cos A = cos II', B e = JLa' sin (J/ + H + v) r 
 
 (3) 
 
 B t = J?a' cos <#>' cos (J^ + ft + v), 
 sin B' = sin II', (7 C = a' sin II' sin /", 
 
 C7 t = a' cos $' cos II' sin 7, 
 
 Substituting these expressions in ' for example : by expansion 
 f ' = / cos II' cos (II -f- v) cos v + r sin II' sin (II -f v) cos 7cos v 
 
 r sin IT cos (II -f v) sin v -f r cos II' sin (II + v) cos </sin vV 
 but 
 
 r cos v = a'(cos ^" e'), / sin -y' = a' cos $' sin ^", 
 whence 
 
 - = ^.X 008 ^^ ~ e ') + ^ sin ^' and similarl y> 
 
 ;' = C (cos ^' - e') + C. sin j^', 
 
7 
 
 III. 
 
 EXPRESSIONS FOR JR Q , 7 , Z . 
 With axes as described it is not difficult to see that 
 
 A 3 ' ~A 3 ' ~A 3 ' 
 
 where A is the distance between the two bodies. We see that 
 A 2 = (f ' rf + 77' -f J" which from (4) and (3) becomes 
 
 A 2 = A - 2# cos (e - E") + C Q coslE" 
 in which we have placed 
 
 ' 2 -f r 2 + 2eWL c = ^ , eV 2 + r^4 c = ^ cos e, 
 
 V / ^ 7-) . ^2 /2 x> 
 
 rA s = ^ sin e, a e = (7 . 
 
 Substituting the values of 7?, 7, Z in (2) we obtain 
 
 " ^ c (cos ,E" - e') -f ^ sin ^ - r 
 == ~ 
 
 (6) 
 
 J9 c (cos E' e) -f B t sin E' 
 
 (1 e' cos E' 
 C 2n C (cos ." e') 4. Q sin _" 
 
 '-i^J, r ~ -SF 
 
 (1 - e' cos E'}dE'. 
 
 These expressions cannot be directly integrated because of the 
 complexity of A in terms of E'. A transformation due to GAUSS 
 ( Werke III, p. 333) makes integration possible by changing the 
 variable E' to a new variable T. 
 
8 
 
 -r = - 
 
 aa" + 
 
 IV. 
 
 GAUSS'S TRANSFORMATION. 
 Let 
 
 E' = a + a' sin T + a" cos T, 
 
 E' = + /3'sin T+ j3" cos T, 
 = 7 + 7' sin J 7 + 7" cos J 7 . 
 Now let these new auxiliaries be subjected to the conditions 
 
 r - 77' = o, 
 
 77" = 0, 
 "' - 7'7" = 0, 
 
 2 rZ f/2 -t Q t ,y " Q" A 
 
 a a a = 1, ap ap a p = U, 
 
 ? 2 ft' /3" = 1, ay a'y a"y" = 0, 
 
 7 2 y' Z y" Z = 1, fty ft'y' ft"y" = 0. 
 
 We now make these auxiliaries such that A 2 may take the form 
 
 (a) ^ 2 A 2 = G - G l sin 2 T+ G 2 cos 2 T. 
 
 Assuming this possible we shall see if real values of the 
 coefficients can be found such that ^V 2 A 2 may take the form 
 
 E' NCOS e 
 
 i^'- JVsin+ CJiNunEJ. 
 
 We now solve (7) for sin T and cos T in terms of E' and 
 observing that 1 = aN cos E' $N sin E' + yN and substi- 
 tuting these values in (a), then equating coefficients with (b) and 
 writing the resulting equations in three groups we have 
 
 I a G a a G l a -j- a" G 2 a" = C 
 SG-a-pGt-a' + P'Gz'a'^Q 
 yG'a y'G^a' + y'Gz-a'^ B Q cos e 
 
 The last two columns are the values of similar equations, the a's 
 being replaced by /3's and 7's respectively. 
 
 (9) 
 
 
 
 sin e 
 
 ,0086 
 
 , sin e 
 
 A 
 
9 
 Whence 
 
 GOL == C Q OL -f yB Q cos e, 
 G/3 = 7# sin e, 
 
 Gy = a^? cos e J9 sin e -f yA Q , 
 GjLi etc., and 6r 2 a", etc., 
 
 give the same expressions with a, /3, 7, replaced by a', y8', 7' and 
 a " " 7 " The coefficients of the a's, /3's and y's are of the same 
 form in each set of equations. The condition that these equations 
 may be consistent is expressed by equating the determinant of the 
 coefficients of a, /3, 7 to zero. This gives when expanded 
 
 #[(# + Q(# - A) + Bl cos 2 e] + Bl(G + <7 ) sin 2 e = 0. 
 
 Thus if N 2 A 2 is to be of the desired form (a), G must satisfy 
 this equation, and since the other two groups give the same equa- 
 tion, except for accents, G l and G 2 must also satisfy this equation. 
 
 Making the variable X and placing 
 
 (10) P^A.-C a P t -Bl-A t C. -P 3 =<7 ^sin'e 
 
 the cubic may be written 
 
 ( C) X 3 - P^ 2 + P 2 X- P 3 = 
 
 GGi and G 2 are then the roots of this equation. The roots can 
 be shown to be real, two positive and one negative. We let them 
 be in order of descending magnitude G, G l and G 2 . 
 
 Multiply numerator and denominator of (6) by N z and the de- 
 nominator will be of the desired form (a). As shown by Gauss 
 NdE' dT and the limits for T are the same as those for E '. 
 Because of the limits terms containing sin T, cos T or sin Tcos T 
 vanish and our transformation gives 
 
 rm V _ R TT z __ i fV r+ r. sin* r+r, 
 
 ^'^'^-2^1 M (G-G lS tf r+<? 
 In which we place 
 
 ( T =/ 7 2 + bay + A/87 - da/3 - la\ 
 
 (12) J r, =yy 2 
 
 =/ 7 " 2 + lay" + Wl" - da"/3" - la" 2 . 
 
10 
 
 
 For S . 
 / = Ae' r 
 
 For UQ. 
 ~^X 
 
 For Z . 
 -Gf 
 
 = AJl + e 2 ) + re' 
 
 ^ c (l + ^ /2 ) 
 
 C c (l + e' 2 ) 
 
 A = A 
 
 5. 
 
 c-. 
 
 d = ^x 
 
 Be 
 
 Gf 
 
 Z = Ae' 
 
 Be' 
 
 Gf 
 
 M ar 
 
 ar 
 
 r* 
 
 It is only in the integration of (11) that Arndt's method differs 
 essentially from that of Hill. Hill's method consists of integration 
 by means of elliptic integrals. The modulus of the elliptic in- 
 tegrals is &=*(G l + G 2 )/(G+ G 2 ). The computation of this 
 modulus necessitates a solution of the cubic ( (7) by approximations 
 for the roots 6r, 6^, 6r 2 . While the solution is not difficult it 
 becomes objectionable because of the number of times it must be 
 made. Arndt's method avoids the solution of this cubic by mak- 
 ing the integration depend upon the integrals of Weierstrass. 
 
 V. 
 
 AKNDT'S MODIFICATION. 
 
 LetTF= r, . dr _ = f ? ^. 
 
 Jo y 6^ G! sin T -\- G 2 cos ^Z 7 Jo ^ 5 
 
 then 
 
 dW fi dT 
 
 N* 
 TdT 
 
 Our expression for V then becomes 
 
11 
 
 Now let sin 2 T= t and (G 1 + #,)/(# + #,) = ^ (modulus), then 
 = when r= 0, = 1 when T= Tr/2 and 
 
 and 
 
 z> 
 
 7 \jf 
 
 where 
 
 X 1 5TF r l dw r l 
 
 Qdt, SG=-P\ Qtdt, j = P Q(l-t)dt 
 U ^TI Jo v ^z Jo 
 
 4(G 
 
 and 
 
 In order to use Weierstrass's function for the integration we 
 make 
 
 
 where s is a new variable and m, 77i 1? m 2 constants which will be 
 disposed of. By differentiation we obtain 
 
 tfdt dt 
 
 ^' 
 
 (k 2 l)dt 
 whence 
 
 , l2 - 
 When t = 0, 3 = (^ x when ^ = 1, s = (r 2 . Now writing 
 
 m = !/((? - G,) and (ff + G 2 )(G - ^)(^ + ^ 2 ) = (7 
 
 and i x 4(s G)(s G^s + G 2 ) = V S^ we have 
 
 dW G, + G n r~ G * ds 
 ^G 
 
 dW 
 
 aw_ G+G 2 .r-* ds 
 
 3G= '20 J Gl ^ 
 
12 
 
 In order that S may be of the Weierstrassian form the second 
 term of the cubic ( (7) must be wanting, that is, each root must be 
 diminished by Pj/3. Hence 
 
 Vsi 
 
 where e^ e 2 , e 3 are the roots of the transformed equation. We 
 then have 
 
 8W G l+ G 2 r*~_^ ds 
 
 , + g, r 
 
 20 1 
 
 dG ' '2C ^ v/^ 
 
 5TF G+ G 2 pa e a 
 'X VI 
 
 5TT Q-Gr'-^j. 
 
 SO,- ^ V~8 
 
 r 
 
 /* 
 
 a^ 2 ~ 2(7 
 
 and 
 
 _ L+_2(e lft > + ^) 
 
 7T 
 
 
 r ds 
 .. 1/ 
 
 We must now obtain the coefficients of o> and 77 in terms of the 
 elements. For this purpose let 
 
 A = (& l + G 2 )T + (G 
 @ = (0, + GJGT + ( 
 then, since e, = G - ^P,, e 2 - G, - JP 15 e s = - <? 2 - 
 
 A and must now be expressed in terms of the coefficients rather 
 than in terms of the roots of ( (7). Let 
 
 and from the symmetric functions of the roots 
 
 ^ GG 2 - G,G 2 = P 2 , 
 
13 
 
 Substituting the values of F, T 1<t T 2 given in (12) we see that 
 the equation for A l reduces to A l f I which by (13) for 
 the three cases gives respectively r, 0, 0. 
 
 By substituting from (12) and reducing with (9) we find 
 
 B l =fA + bB cos e + hB Q sin - I <7 , 
 whence from (13) 
 
 f = - A Q (A e e'+ r) + B cos {eV 
 (15) 
 
 B* A 9 C.e'+ -# cos e(7 c (l + e 2 ) + B Q O 8 sin e - <7 c <7/. 
 
 For 5^ it is preferable to substitute the primitive values, whence 
 we obtain 
 
 (16) J?f = - Jra' cos 2 ^ sin 2 ,/sin 2(11 + t>) - ^'^c- 
 
 For reducing C l we multiply the equations Got 2 0^+ G 2 a" 2 =z C Q 
 and G + #! - ^ 2 = P l by a' 2 -f a" 2 - a 2 = 1 and 
 GG 1 GG 2 0^2= P 2 respectively and add the products. 
 We thus have 
 
 a'G.G, - a 2 GG 2 -f a'GG, = P 2 -f P l C + a / (7 2 + a^ - 2 (7 2 . 
 
 The last three terms can be replaced by known quantities. Mul- 
 tiplying 
 
 Go. = C a + B Q cos 67 by Ga , 
 
 ^ cos 7 Gjot, 
 
 and adding the products we obtain 
 
 ~ ^ 2 2 = Cl - ^ 2 cos 2 e. 
 
 Dividing by G&G = P 3 we get 
 
 a 2 a <t" 2 .ggin a 6 
 
Similarly we obtain 
 
 14 
 
 dff a"/3" 1 ^ , 
 
 -rr -f -7T- = ^ -#o sm e cos e, 
 
 7 a 7 ay 
 3y Py . /3" 7 " 
 
 cos e 
 
 sin. 
 
 -Cosine 
 
 The three terms in C^ occur in the form 
 
 V b + ^h a/3 d -I 
 b h G* ~ G ~ G 
 
 The other two quantities are given by the same equation with ap- 
 propriate accents. Substituting in the equation we find 
 
 - P s Ci = B Q sin e(cLS cos e - IB Q sin - hC Q ). 
 Substituting from (13) for each case we find 
 
 
 Ar'e'ABa' 2 cos <' cos (.4' - H), 
 
 To obtain A we consider the identity (v. HalpJien FonGtions 
 Uiptiques ch. VIII). 
 
 3 
 
 - r i\ r 2 
 
 
 111 
 
 
 0i -0 2 
 
 
 
 111 
 (? ^ ""^" 2 
 
 = 
 
 111 
 
 
 /^ /"* /^ 
 
 tr Cr r Cr 2 
 
 
 Multiply by the equation GG 1 G 2 P B using a factor in each 
 column of the second determinant. Subtract the last column 
 from each of the first two and upon expanding we find it equal to 
 + C. In the first determinant multiply the first column by 
 
15 
 
 _ ( Q^ _j_ y and multiply the last two columns by ( G + 6r 2 ) and 
 (G l G) respectively and add them to the first column. We 
 thus find for this determinant thus modified : 
 
 -A 
 
 1 
 
 A 
 
 
 
 r, 
 
 ff, 
 
 i 
 
 r 2 
 
 -i 
 i 
 
 G 1+ G 2 
 
 Our identity when expanded then becomes 
 
 - B^P, - 9P 3 ) - 
 Now let 
 (18) P 1 P t -9P,- P , P?-3P 2 
 
 We then obtain for the three cases 
 
 J - 3P a ). 
 
 (19) H 
 
 If we write the transformed cubic in the usual manner 
 4 S 3 _ g^s g 3 = 0, we have the following equations for the roots. 
 
 e 2 + e 3 = 0, 
 
 4( ,-,- 
 
 P 2 
 
 If g z and g 3 are the invariants of the cubic whose roots are 6r, 
 the discriminant is g\ 27#* and C7 2 , i. e., the product of the 
 squared differences of the roots will be 
 
 (21) C". 
 
 (22) 
 
 g is the absolute invariant. 
 
16 
 
 Again for 
 
 G 
 
 we consider 
 
 an identity 
 
 
 G 2 T 2 
 
 
 1 
 
 1 
 
 1 
 
 -G. 
 1 
 
 - 
 
 1 
 
 G 2 +6 
 
 1 
 ^ 
 
 1 
 G, 
 
 G 
 
 ~ 0, 
 
 G, 
 
 .-A, A 
 P, 3 
 
 -p.- 
 
 Multiply this equation by GG t G 2 = P 3 , using a factor in each 
 column of the second determinant, the negative sign being used in 
 the last column. Next subtract the last column from each of the 
 first two ; the second determinant when thus treated and expanded 
 will be found equal to 2\( G l + 6r 2 ). In the first determinant 
 multiply the columns respectively by ( G^ + 6r 2 ), ( G -f 6^), and 
 (G l G). Subtract the last two columns from the first. The 
 determinant thus treated and expanded will become j(G l + G 2 ). 
 Our identity then becomes 2\O = A(P 1 P 2 9P 3 ) + C(P^ 
 Whence we have for the three components 
 
 (23) 
 
 and for the components we have finally from (14) 
 2 FA*/ P, 
 
 (24) 
 
 2 
 
 P 
 
 The elliptic integrals w and 77 are computed by the method of 
 H. BRUNS.* Writing for the moment ^o> a = e a we know that 
 
 {K 4- u) - e a }(s - e a ) = (e a - e^(e a - e y ). 
 
 *"Ueber die Perioden der elliptischen Integrate," Math. Ann., Bd. 27. 
 
17 
 
 Then 
 / = 
 
 ( s - 
 
 Let 
 
 . s = 
 
 5 
 
 ds = 7-7 
 
 , e e 2 g 3 
 
 e i H ir , 
 
 When 
 
 5 = e a , s' = 
 
 ds 
 
 5 = e a , s = <? 3 , s = c s , s = e 2 , 
 
 * ds r* a'da' 
 
 ^ + ->7S"JL^ 
 
 If now ^ 2 and ^ 3 are given functions of f , we obtain, by differ- 
 entiating the equations for co and 77, 
 
 d< p (?; + &),, ^_ f'(y; + y 3 ) ,. 
 
 df-J., 2^i rfs ' d|~ -J e , ' 
 where 
 
 / d $2 A ' d ff* 
 
 ^-? and ^ 3 = ^. 
 But we have the identities 
 
 ,- 6^-63) (.-,) 
 
18 
 
 and introducing the integral I 
 
 sds 
 
 whence 
 
 (25) ~=-Pr,-Q a> , | 
 
 where 
 
 t 
 ' 
 
 or expressed in terms of the symmetric functions of the roots 
 
 - - 86^; + 
 
 If now we substitute for &> and 97 in (25) respectively 
 
 10 
 
 the coefficients of the differential equations can be expressed by g 
 and its derivatives only and we obtain 
 
 n 
 
 -i) 1 
 
 and now for g'd% we write dg. 
 
 Eliminating H and H respectively from these equations we get 
 
 . dl 55 
 
19 
 
 These equations are of the form of the differential equation 
 whose solution is the hypergeometric series, i. e., 
 
 = x(x- 1) + [>(a+ fl + l)-7] + apy 
 
 where a = $, ft = T 6 2 , 7 = f , and a = T \, = ^ 7 = J. One 
 of the 24 solutions of this equation is 
 
 y 
 
 O) = 
 
 To determine the constants (7, (7' we place 
 = 1, 0r 3 = 1, then # 2 = 3, and e^l, e 2 = e 3 = - }, t? = e i~ e * = Q, 
 
 and since 
 
 1 C d6 
 
 CO = - =^:^ - ^ 
 
 l/ 6i e 3 Jo VI /fc 2 si 
 
 TT 
 
 
 
 sin 2 <^ 1/6 
 
 l e 3 1/24 
 
 since when g = 1, (</ !)/# = 0, and F 1 
 
 .-. ^ ==^d I t> = || . , V^ ==^J I "t - = 4d i , 
 
 1/6 iX!2 1/24 i!/1728 
 
 whence 
 
 (26) -j^ 
 
 where JP, and ^ are the two hypergeometric series 
 
 For computation we first solve the triangle described in part II, 
 for the preliminary constants, then apply the formulae in the fol- 
 lowing order 3, 5, 10, 18, 20, 22, 21, 15, 16, 17, 19, 23, 26, 24. 
 The values of F m and F^ required in 26 may be taken from the 
 
20 
 
 table given by Arndt or they may be computed directly. The 
 argument of this table is x = (g l)/gr. 
 We now compute the following expressions : 
 
 HU Z Q sin u, H { Z Q cos u, H e 7? sin v -f 7 (cos v + cos E), 
 
 1 + sin 
 
 r 
 
 1 + 5(1^ 
 
 We now take the mean value of each quantity, which we repre- 
 sent by (fffi), etc., and obtain the variation of the elements from 
 the equations : m > 
 
 = 1 +ra' 
 
 sin (Sfl) = B sec <(^T ), (Si) = B sec 
 (8e) = B cos <t>(ff e ), e (^ w i) = -S cos 
 
 cos 
 
 (SJQ + 2 sin 2 ^^^) 4. 2 sin 2 1 (an). 
 
 X and TT are the mean longitude and the longitude of perihelion 
 measured from a fixed epoch. 
 
 VI. 
 
 COMPUTATION. 
 Saturn upon Mars. 
 
 The elements taken from Dr. G. W. Hill's " New Theory of 
 of Jupiter and Saturn " are 
 
 Mars p. 192. Saturn pp. 19,558. 
 
 7r=33317'51".74 TT'= 90 6'4r.37 
 
 i= 1 51 2.24 i'= 2 2940.19 
 
 H = 48 23 54.59 ft' = 112 20 49 .05 
 
 e = 0.09326803 e = 0.05606025 
 
 P = 689050".784 p' = 43996"21506 
 
 log a = 0.1828971 log a = 0.9794956 
 
 m 1/3093500 m = 1/3501.6 
 Epoch 1850.0 G.M.T. 
 
21 
 
 The preliminary constants are found to be : 
 
 n = 17617'59".42 
 IT = 293 4 38 .78 
 
 K= 108 35 57 .73 
 1C = 44 41 13 .54 
 
 J= 2 2152 .11 
 
 A = 0.99927955 
 # = 0.9998659 
 ^' = 6654'17".84 
 B' = 66 56 24 .55 
 
 The values of the various quantities given by the formulae are 
 given in the table below. The residual in Innes's test equation 
 was found to be 0.000,000,000,2. 2 t and 2 2 are the sums of 
 the values for the odd and even points of division 
 
 E 
 
 log r 
 
 V 
 
 log A. 
 
 log^ c 
 
 log A. 
 
 logJ? g 
 
 
 
 0.1403760 
 
 o o' o!6o 
 
 0.6331692* 
 
 0.9298510 
 
 0.9295583* 
 
 0.63221 33 n 
 
 30 
 
 0.1463201 
 
 32 47 24.62 
 
 9.9980540 
 
 0.9768010 
 
 0.97634 55 n 
 
 0.0001597 
 
 60 
 
 0.1621568 
 
 64 44 46.64 
 
 0.7680496 
 
 0.8760042 
 
 0.8753687* 
 
 0.7679641 
 
 90 
 
 0.1828971 
 
 95 21 5.91 
 
 0.9480287 
 
 0.5421705 
 
 0.5410633 n 
 
 0.9477062 
 
 120 
 
 0.2026919 
 
 124 31 47.16 
 
 0.9752134 
 
 0.1081758 n 
 
 0.1097082 
 
 0.9747498 
 
 150 
 
 0.2166314 
 
 152 34 23.40 
 
 0.8883581 
 
 0.7460753* 
 
 0.7460180 
 
 0.8877384 
 
 180 
 
 0.2216237 
 
 180 0.00 
 
 0.6331692 
 
 0.9298510n 
 
 0.9295583 
 
 0.6322133 
 
 210 
 
 0.2166314 
 
 207 25 36.60 
 
 9.0215124 n 
 
 0.9791564 n 
 
 0.9787259 
 
 9.0446347* 
 
 240 
 
 0.2026919 
 
 235 28 12.84 
 
 0.6602801 n 
 
 0.9223582 n 
 
 0.9217848 
 
 0.6603403* 
 
 270 
 
 0.1828971 
 
 264 38 54.09 
 
 0.9069056 n 
 
 0.7051686 n 
 
 0.7043164 
 
 0.9066458 n 
 
 300 
 
 0.1621568 
 
 295 15 13.36 
 
 0.9790254 n 
 
 9.4089485* 
 
 9.3984933 
 
 0.9786051 n 
 
 330 
 
 0.1463201 
 
 327 12 35.38 
 
 0.9148835 n 
 
 0.6835510 
 
 0.6835778 n 
 
 0.9142995 n 
 
 ~sT 
 
 1.0916971 
 
 900 0.00 
 
 
 
 
 
 2, 
 
 1.0916972 
 
 1080 0.00 
 
 
 
 
 
 E 
 
 10g5 
 
 log^ 
 
 p 
 
 \ 
 
 g 
 
 logC" 
 
 
 30 
 
 1.0710308 
 1.1693425 
 
 1.9648845 
 1.9689849 
 
 10682.6599 
 18227.1670 
 
 8117.2267 
 8041.3776 
 
 1.0027367 
 1.0056470 
 
 8.3349592 
 8.6360598 
 
 60 
 
 1.2416284 
 
 1.9733790 
 
 26314.7192 
 
 7960.4005 
 
 1.0092812 
 
 8.8370927 
 
 90 
 120 
 
 1.2868582 
 1.3068094 
 
 1.9769311 
 1.9787352 
 
 32935.6960 
 36424.3060 
 
 7895.1366 
 
 7862.0877 
 
 1.0129818 
 1.0155661 
 
 8.9705083 
 9.0427814 
 
 150 
 
 1.3022786 
 
 1.9783135 
 
 35792.7330 
 
 7869.9176 
 
 1.0157721 
 
 9.0497000 
 
 180 
 
 1.2725893 
 
 1.9757393 
 
 31049.5750 
 
 7917.3434 
 
 1.0131888 
 
 8.9809495 
 
 210 
 240 
 
 1.2158096 
 1.1302863 
 
 1.9716558 
 1.9671494 
 
 23362.5420 
 14847.5260 
 
 7992.6957 
 8076.0143 
 
 1.0087905 
 1.0043632 
 
 8.8189886 
 8.5302017 
 
 270 
 300 
 
 1.0231950 
 0.9419157 
 
 1.9634684 
 1.9616475 
 
 7941.6038 
 4595.4802 
 
 8144.1096 
 8177.6560 
 
 1.0013537 
 1.0002558 
 
 8.0341618 
 7.3163722 
 
 330 
 
 0.9692790 
 
 1.9621814 
 
 5654.5662 
 
 8167.4836 
 
 1.0008465 
 
 7.8342202 
 
 1 
 
 
 11.8215349 
 11.8215351 
 
 123914.27 
 123914.31 
 
 48110.729 
 48110.721 
 
 6.0453918 
 6.0453916 
 
 
22 
 
 E 
 
 lOgPstf/ 
 
 log-P.Ci* 
 
 B* 
 
 B u 
 
 Bp 
 
 A*C 
 
 
 
 1.9169034 
 
 0.5318556 
 
 1.71845 
 
 .89671773 
 
 0.3721019 
 
 1273741.5 
 
 30 
 
 1.9755785 
 
 0.5328775 
 
 3.01775 
 
 -1.13441580 
 
 2.5062115 
 
 2172708.2 
 
 60 
 
 1.9062750 
 
 0.2070736 
 
 -4.62374 
 
 .98414281 
 
 4.7095692 
 
 3211393.9 
 
 90 
 
 1.6134503 
 
 8.6889828 n 
 
 6.08172 
 
 .44676768 
 
 5.6474343 
 
 4168957.1 
 
 120 
 
 1.2216849 n 
 
 9.5472262 
 
 6.93032 
 
 0.29124699 
 
 5.0684299 
 
 4791819.1 
 
 150 
 
 1.8858736,, 
 
 0.4342181 
 
 6.88475 
 
 0.95971490 
 
 3.1277779 
 
 4857869.0 
 
 180 
 
 2.0793986 n 
 
 0.6943511 
 
 5.96030 
 
 1.34015180 
 
 0.3456203 
 
 4287900.0 
 
 210 
 
 2.1185815 n 
 
 0.6960869 
 
 4.47493 
 
 1.35536890 
 
 2.5324875 
 
 3228683.9 
 
 240 
 
 2.0337615 n 
 
 0.4411836 
 
 2.90359 
 
 1.07265140 
 
 4.7354355 
 
 2023242.3 
 
 270 
 
 1.7767034 
 
 9.5893663 
 
 1.66622 
 
 0.62356451 
 
 5.6730943 
 
 1059477.9 
 
 300 
 
 0.4293996 
 
 8.6103629 W 
 
 1.00508 
 
 0.10695994 
 
 5.0942969 
 
 597208.9 
 
 330 
 
 1.6828108 
 
 0.2039025 
 
 1.01624 
 
 .42731476 
 
 3.1540537 
 
 697610.1 
 
 2, 
 
 
 
 23.14148 
 
 0.9301496 
 
 0.0782150 
 
 16185305.7 
 
 
 
 
 23.14161 
 
 0.9301500 
 
 0.0782128 
 
 16185306.2 
 
 E 
 
 
 30 
 60 
 90 
 120 
 150 
 180 
 
 A"C- 
 
 A Z C 
 
 *$ 
 
 & u 
 log - c - 
 
 ^ 
 
 1,000 R 
 
 1331148.8 
 1499656.7 
 1257141.7 
 633680.8 
 - 251362.3 
 -1175895.1 
 1859502.8 
 
 51269.99 
 100538.88 
 149578.11 
 185230.51 
 -190157.71 
 154729.51 
 - 89067.22 
 
 8.0562601 
 8.1210086 
 8.1877509 
 8.2373935 
 8.2639876 
 8.2663024 
 8.2448728 
 
 7.5893686 
 7.5702475 
 7.4530372 
 7.1237659 
 6.6740144 M 
 7.3428058 
 7.5509401 
 
 5.9407458 n 
 6.8638621 n 
 7.095934U 
 7.1721544 n 
 7.1355044* 
 6.9590236 n 
 6.3943817n 
 
 1.7097689 
 1.7745204 
 1.9262896 
 2.1375744 
 2.3569481 
 2.5168682 
 2.5606371 
 
 210 
 240 
 
 2068748.7 
 1729849.5 
 
 20232.80 
 25701.71 
 
 8.2024945 
 8.1483792 
 
 7.6367383 n 
 7.6523844 n 
 
 6.6338873 
 6.9773713 
 
 2.4720554 
 2.2843136 
 
 270 
 
 969088.3 
 
 38725.85 
 
 8.1153154 
 
 7.6286630 n 
 
 7.0861514 
 
 2.0608640 
 
 300 
 330 
 
 I 
 
 43468.5 
 784500.8 
 
 24077.58 
 
 8288.07 
 
 8.2046672 
 8.0486528 
 
 6.7557748 n 
 7.5910007 
 
 7.1006984 
 7.7301623 
 
 1.8653206 
 1.7413404 
 
 1295892.6 
 1295893.9 
 
 430293.74 
 430293.93 
 
 
 
 
 12.70328 
 12.70322 
 
 E 
 
 1000 U 
 
 1000 Z 
 
 1000 # n 
 
 1000 H t 
 
 1000 H u 
 
 1000 He 
 
 -500 H Ml 
 
 
 
 +.0099129364 
 
 +.01219913 
 
 .01178899 
 
 +.00313663 
 
 1.7097689 
 
 +0.01982587 
 
 1.5503021 
 
 30 
 
 +.0070243377 
 
 .10023809 
 
 +.06747523 
 
 .07412671 
 
 1.4844357 
 
 +0.97300477 
 
 1.6311880 
 
 60 
 
 +.0054967025 
 
 -.20386986 
 
 +.03664317 
 
 .20054972 
 
 0.8120554 
 
 +1.74728269 
 
 1.8364587 
 
 90 
 
 +.0055926247 
 
 -.27213016 
 
 .09419268 
 
 .25530876 
 
 +0.2105525 
 
 +2.12773530 
 
 2.1375744 
 
 120 
 
 +.0032050993 
 
 .27013335 
 
 .20519340 
 
 .17569210 
 
 +1.3414279 
 
 +1.93830949 
 
 2.4668628 
 
 150 
 
 .0049176511 
 
 .17936971 
 
 .17509923 
 
 .03890714 
 
 +2.2292371 
 
 +1.16793251 
 
 2.7201623 
 
 180 
 
 .016050410 
 
 .02369097 
 
 .02289447 
 
 +.00609141 
 
 +2.5606371 
 
 +0.03210082 
 
 2.7994626 
 
 210 
 
 .022805479 
 
 +.13539885 
 
 +.10010378 
 
 .09117062 
 
 +2.2161522 
 
 1.09867500 
 
 2.6717296 
 
 240 
 
 .019925518 
 
 +.23709997 
 
 +.07965469 
 
 --.22331936 
 
 + 1.3285740 
 
 1.86063280 
 
 2.3908407 
 
 270 
 
 .0090535750 
 
 +.25685432 
 
 .04260349 
 
 .25329642 
 
 +0.2103199 
 
 2.05103630 
 
 2.0608640 
 
 300 
 
 +.0029944062 
 
 +.20818148 
 
 .13424167 
 
 .15911853 
 
 0.8010985 
 
 1.68427280 
 
 1.7783332 
 
 330 
 
 +.0097886250 
 
 +.11927107 
 
 .11350343 
 
 -.03664085 
 
 1.4740911 
 
 0.92634119 
 
 1.6006880 
 
 2 i 
 
 .014366784 
 
 .04021460 
 
 .25782067 
 
 .74945050 
 
 +1.907705 
 
 +0.19261327 
 
 12.8222601 
 
 *\ 
 
 .014371118 
 
 .04021372 
 
 .25781983 
 
 .74945177 
 
 +1.907735 
 
 +0.19262010 
 
 12.8222063 
 
23 
 
 The resulting values are 
 
 (&) 
 e(S7T) 
 (SZ,) 
 
 sn 
 
 - ".024687315, 
 ".0062897022, 
 ".062294586, 
 ".93075550, 
 
 ".062281786, 
 
 - ".83828575, 
 
 - ".0084927392. 
 
 For comparison we give the results of Newcomb and of Lever- 
 rier and those obtained by applying Hill's method.* Leverrier's 
 values are reduced to the value of m here adopted. 
 
 Newcomb. 
 
 Leverrier. 
 
 Method of Hill. Method of Arndt. 
 
 (*) 
 
 (<*) 
 
 (6L) 
 
 + 0.00629 
 
 + 0.00628 
 
 + 0.0062891 
 
 + 0.0062897 
 
 + 0.06226 
 
 + 0.06226 
 
 + 0.0622814 
 
 4- 0.0622817 
 
 0.00849 
 
 0.00852 
 
 0.0084927 
 
 0.0084927 
 
 0.02468 
 
 0.02467 
 
 0.0246873 
 
 0.0246873 
 
 
 
 0.838 
 
 0.8382821 
 
 0.8382857 
 
 CONCLUSION. 
 
 The computation apparently shows that this method is some- 
 what less accurate than that of Hill. The cause lies in the fact 
 that the formulae involve differences of almost equal quantities. 
 This is true, in this problem at least, in the formulae for .Z?f, C 2 
 and U Q . For example, the formula for C 2 (21) contains the factor 
 (g 1) ; a glance at the values of g given in the table shows that 
 this factor becomes so small that its logarithm, and consequently 
 that of (7 2 , becomes quite inaccurate in its last places. Whether 
 these conditions are accidental to the problem is not evident. In 
 spite of the greater mathematical elegance of the treatment by 
 Arndt, it is the opinion of the author that the additional accuracy 
 secured by applying Hill's method would repay the extra com- 
 putation. 
 
 *A. J., No, 574.