14 GENERAL INTEGRALS OP PLANETARY MOTION. 



From these expressions it may be shown that 



(hJj)-=0 (20) 



in the same way that we found ((/.j., cij) = 0. 



We have next to consider the combinations of. the form (a^, Ij), for which the 

 expression is 



(ajj) = ^ ^Jil^ — Jp^-^^ ' »_ etc. 1^ 



The terms which do not contain t as a factor are found to be 



_«,«■,{ ft«), g + te% *]**i' } cos (i^, - «) 



-i s^s: { am. f ; + (A/.0, ^1^- } COS (^, - ^.). 



S' having the meaning given on page 12. 



The only non-periodic terms in this expression will be those in which [.i = v, and 

 these terms reduce to 



e„ f .jj 61c , .^6(1)^) , 1 .-,,-,,61^ , . ., 6(b'l0 \ 



__o^\6{ijhJ^ 60?'^] 



~ \ 6a, ^^ 6a, f 



or, by putting 



Cj = S' ^ijhP + kjW] (21) 



we have 



(a„Z,)=_|^. (22) 



6o,, 



These expressions are now co be substituted in the differential equations repre- 

 sented by (15), which will then divide into two classes according as the derivative 



of il is taken with respect to ?i, Zo or lg„, or Avith respect to a^, a, or 



(isn- Having regard to equation (20) we find those of the first class to be of the 

 form 



^'^' "^-^ Tt + ^^^■' "^^ rf7 + + ^''' """'^ -dt-m,~.^.W6ij- 



If, in the first member, we substitute for the coefiicients their values (22), noticing 

 that 



{Jp «'^) = — (%' ^j)' 

 and in the second member put for brevity , 



6lj ^' \ 6f 6lj "^ 6f 6lj ^ 6f 6lj / — ^' 



the difi'erential equation reduces to 



6cj day do J dn,,^ 6cj da^^ ^ 



da^ ~dt ^ da^ldt '^ + ^ "^ — i' 



or 



|=n„ (23) 



