GENERAL INTEGRALS OF PLANETARY MOTION. 25 



In (43) we have 



and, integrating. 



OCj j = l OCj 



= _^J^ '^J'^l^i^^ 



2 '^-/Isiniv; 



(44) 



Avhich, for brevity, we may represent by 



h'ki = /S',Xj sin N, 

 putting 



7i"; , •'■=^" 7t',- ^6i 

 y=i 6cj 



In adding the effect of the perturbations hci to ^, yj, and ^', we are to vary only 

 k, the expressions for S'^, etc., being 



S^=S.A hh cos i\^— Z; sin N{iMi + ^35A2 + + Hn^^zn) \ 



Syi=:Su I bh sin N-\- k cos N^i^S^ii -j- igi^/ls + -\- isn^^sn) \ 



^^ = ^„ I bJc:sinN'^k'cosNXj\8^^-]-j,h^,+ +J3JW \ 



We are to put in these expressions 





and the values of ^;i in (44). We thus find 



(45) 







. . + ianL-Snl \ COS (N^ + N,) 

 ■ ■ + isn^sn), I COS (iV^ — iV;,) 



2, (^ ^ ). + ^^ (^■li:, + i,L, + + *,„X3J, } sin (^. + iV^,) 



2, (^ 1^ ), - A;^ (*,X, + *,i, + + i,„L,„l I sin (iV^ — i\^0 



2. {^^l-y + k',0\L, +j,L, + +j;A„)„ I sin {N', + iV,) 



2, (^ ^'),-yi-'^(yiZ, +.y;x, + +i.X3j, I sin (^; - n,) 



Since, in JV^ we have 2*= 1, 

 while in N, " " 2* = 0, 



November, 1874. 



