7] THEORY OF JUPITER'S SATELLITES. 175 



But if we omit the disturbances due to eccentricity and also those 

 due to the square of the disturbing force, we may write 



1 dR I S rn _. m' dQ 



-T- = - -ni[l + 3co82(0-Z)] + 2- -V*, 

 r dr 2 Z>" L a da 



1 a cos (I I') 



where Q - -V- 



I = nt + e, l' = n't + f'. 



To satisfy these equations, assume 



r = [1 - c cos 2(l-L)- 2<7, ; cos i (I - /')], 

 6= l + ksm2(l-L) + 2,g i smi(l-r), 

 where i includes all positive integers. 



Then neglecting squares and products of the coefficients a it g it c, k, 

 and also ,- or M in places where it will afterwards appear that its 



retention would not modify the result to the order we are considering, 

 we have 



- ~ = 4n*c cos 2(1- L) + Sr (n - nj a, cos i (I - /') 



= - n- - ri>k cos 2(l-L)-2n(n- n') % 4 cos i (I - 1') 

 + ^ = - 3 [1 + 3c cos 2 (I - L) + 32a f cos i (I - 1')'] 



I I Cv 



+ - e [1 + 5c cos 2(l-L) + oSa^ cos i (I - 1')] 

 ct 



= -,[1 +/+ ( 3 + 5 /) c cos 2 (^ - X) + (3 + 5/) Sa ; cos i (I - Z')] 



where f= v 



Also 



5r= - 4 2 yfc sin 2(l-L)- Sr (n - n') s ^i sin i (I - 1'), 



_ _ n ta . sn 4 _ 



Ct-C 



