GENERAL INTEGRALS OF PLANETARY MOTION. 



'l/mo y'm-i^ -[/m^ -y/ni^ 



-j/??i ' -j/m ' |/m ' • • • • ^^^j 



4^^, "'^, 0, 0, (10) 



•^20 0(21 0^22 rv f\ 



■y/Wo' "l/Wi' -1/77*2' '• • • • 



^710 CC?il 0f.n2 ^«.?j 



V^^o' l/«^i' l/"'2' • • • • -j/mj 

 Then a,in will be determined by the condition 



aL _, "^« 1 



— ■'■5 



«?„ m 



while all the other coefficients in the bottom line will be determined by the condi- 

 tion 



vf-^= + - — -^ = 0. 



Taking the line next the bottom the diagonal coefficient will be determined by the 



equation 



2 [ 2 



Q'ra, n—l ~r (^n—l, n~l 1 l^n—i. -1 



while the remaining coefficients of the form a„_,^ ^ Avill be given by the equations 



The general values of the coefficients to which we are thus led may be expressed 

 in the following way : put 



^7; = ^0 + W?i + . . . . OTi, 



by which 711 will become ^„. Also, suppose 



y f i,- f7j_i 



We shall then have 



aji = — Vjmi. . . . {i<j). 

 It is easy to prove that the coefficients thus formed fulfil the required conditions. 



If we substitute these values of the coefficients in the expressions for ^1 and ^2? 

 they become 



^1 — :-7-=-= (a?! — ^0) 



y »7o + ^1 



^2 = / — ^ ( ("^o + '"1) a;., — «?! .J?! — Too ^'0 )• 



