GENERAL INTEGRALS OF PLANETARY MOTION. 21 



This expression reduces to the form S H cos N, where in each value of iV^ we 

 have 



2i = 0. 



In this expression it may be worth while to give the complete value of H corre- 

 sponding to any value of N. The value of the latter is completely determined by 

 the indices i'l, i^, etc., which multiply \, /I2, etc., in its expression. Let then 



represent the value of N for which we wish to find the corresponding value of 



Hj{i^id-i %„) by means of (35). The required term will be found by taking 



in (35) all combinations of v and j^l for which we have 



N,~N^=^ iV, 



N\ — N'^ = N, 



or N\ + N'^ = N. 



Let as represent the combination of indices v in N, by A'l, A'j, etc., and those in 

 -^'v byji, ^25 etc., so that we have 



N, — ^1X1 + ^<2^2 + + f'3«^3«> 



N\ ^j,\ +j^-k, + +y3„^3.. 



Then, in order that the sum or difference of these angles and of iV^ may make iV, 

 according to the formulae just written, we must have 



^1^ = (f'l — *'l)^l + (^'2 ^'2)^2 + + (^^3,1 ^sJ^Sn, 



and 

 or 



N'^ = (i\ -yOX, + Ci2— ^;)?.2 + ..'...+ {hn—hn)hn' 



For the corresponding coefficients of the time b, we have 



^^ = (."1 — *l)^l + (^^2 — h)^ + + Uhn hn)hn 



b'f. ± (il — h)h d= (J2 — Qh ± zkijsn — hn)hn- 



Aifecting k and k' with the proper indices, as explained in § 4, the part of the 

 coefficient Ej{ii, h %„) corresponding to any one value of the angle N„ will be 



i — n ■ 



2 ^jki{^i,[X2, ) hi^h — ii,^i2 — h, ) V 



+ I f^Jjf^iUl^j2, Wf. i yijx—h,j-2—H, )—k'i{h—jl, h—J2, )> 



where the values of &^ andS'^are those just given. The complete value of ^(ij, i^, . . . .) 

 will be found by taking the sum of all the terms which we can form by giving to 



l«i, ij.2, etc., j\, j„, y3„, in these expressions, all admissible combinations of values, 



that is, the complete expression will be given by writing before the first line the 

 symbols 



i«i=OC /«2=0C ^311 =oc 



2 2 2 



/«i = — OC |i.t2 = — OC i«3,. = — oc 



