GENERAL INTEGRALS OF PLANETARY MOTION. 23 



resulting values of q and li in the expressions (12). Representing the perturba- 

 tions of each quantity by the sign h, we shall have to increase each value of /L by 

 the quantity 



h%i = Ui + thh 



We here have the time t outside the signs sin or cos in both Ui, from the integra- 

 tion of (38), and in thhi. We must next find the sum of the terms thus introduced 

 into h'ki. Differentiating this expression we have 



We have now to form the sum of the terms in the second member of this equation 

 which are multiplied by t. Beginning with the second, we have, omitting the in- 

 dex of h 



dh 6b dcy 1^ 6h dc.^ 



dt dc-^ dt dc^ dt 



Substituting for ~ their values in (37), this equation becomes 



c76 

 di 



„ { ^, db , ^, db , , r <56 1 . ,7- 



which, after multiplying by t, is to be added to the last member of (38). But it 

 will be more convenient, instead of using Ji and h'" in these expressions, to retain 



the expressions —^, — ^, and — - in their present analytical form. Eepresenting them, 

 dt^ df df 



for brevity, by ^", yi'\ and ^", the equations (23) and (24) become 



dt- ^-,Ji^^^ + ^i^ + ^*^i (40) 



dt dcj ^ i = i n '(9c,. ^ ^^ 'dCj ^ ^ 'dcj j 



If in the first of these equations we substitute for the derivatives their values in 

 (34)' and (36), it becomes 



^ = _ Sf I ^A - 2 {em I sin iV + S {v"iij h) COS iV + 2 (^'\jj h\) cos N'. 



Substituting in the first of the above expressions for — , we have 



dt 



dh a S . dh , . db , , . db 



dt 



V i . dh db . . . db ] . . ^^ 



(41) 



