XVIII.] ON HILL'S METHOD OF TREATING THE LUNAR THEORY. 83 



Also multiplying the differential equations for Sx, Sy by sin tf>, cos <f), 

 respectively and adding 



. . d*8x . d8y . / . dSx . 



~ sm - + cos - + 2n cos + dt + sm 



s , . . . 



y cos * to + sm ^ +a cos * J 



X sin (f) + Y cos <f). 



Substitute and we find 



d*w ^ dv d<b /dd>\- d'<h , fdv dd> 



I O _ '_.)/!/ ' I _1_ 01 __ I. 4. QTI' I _ _ 0/1 ' 



dr ^ z dtdt dt) v df + Zn (dt w dt 



. . ,, , . 



v , sm rf> cos <b + -7 7- (cos" d> snr <f>) + - 1 - sm d> cos d> 

 2 ^ dy- 



sin <f> cos 



X sin ( + F cos <f). 

 Now we have seen 



dv v dV 



jl= i Tr-37 

 dt V dt 



Substitute for 2+w' on the left. 

 We get 



= 01 



+ w \ J~- 

 L "* 



X sin (f> + Y cos 



112 



