7] THEORY OF JUPITER'S SATELLITES. 147 



Substitute the above coefficients, and divide the first equation by 

 <fa? and the second by (fa ; the equations then become 



+ cos 2qt ( - 3/c 2 + 1 2/V - ~ /V) + cos 40 (g / 

 [sin (2/c - 6/< C + 4 | / V) + sin 30 ( - ^ / 



1 rf 2 /8z\ /8z\ |~ ... 71 , , 413 ,., , 



? a? W " - W L 1 ~ 4/c " + T fc ~ -24 /v 



+ cos 20 7/c 2 - /V + li 5 / v + cos ^qt-~ f v 



, 1261 -. , , 13493 ft \ . I 1171 

 + sm - 



o 



Now suppose a term in - 2 to be ecos(p + /3), for which we shall 



(/ 



simply write e cos pt. 



If we substitute this term for in the second equation and at 



a 



first omit all the terms of the equation containing / as a factor, we have 



1 d? /8z\ 82 3 r . / % . 



-j- I ) H = See sin at cos pt = ~ ce [sin (q p) t + sin 



q at \Ct/ ci 2 



Hence 



a 2~Lp(2q-p)' P&q+P) 



Now again substitute this value of z/a in the terms which contain 



192 



