138 THEORY OF JUPITER'S SATELLITES. [7 



Thus we have eliminated one coordinate, and have obtained a pair 

 of simultaneous equations between r and z ; we proceed to integrate these 

 equations. To integrate completely we must introduce four new arbitrary 

 constants. We shall first consider the case in which two only are present, 

 namely the inclination, and the longitude of the node. 



Assume 



2 = ac [sin (qt + y] + c l sin 3 (qt + y) + c i sin 5 (qt + y)], 



where c, y determine the inclination and the position of the node, while 

 a is a third arbitrary which, it will appear, is involved in C already 

 introduced. For brevity we shall omit the constant y, and write 



Considering y as a small quantity of the first order, we shall find 

 that rt i; c,, are of the first order and a 2 , c 2 , of the second. 



Substitute for r and z, and we find to the second order 

 r a 2 1 2a, cos Iqt 2 2 cos 4qt + -/(!+ cos 4qt) , 



I d~ 



- (r 2 ) a 2 r4o 2 a, cos 2qt + (I6q-a 1 2o ,'V) cos 



O ^/*- \ / Li 1 Z \ i * *lt 



z* etc 



-= - fl +a, cos 2gt + a,cos ^qt], 

 r a 



vtf v(? 



tl 5 / 1 5\ / 5\ "1 



---,+ -o + c ^ + o a > cos 2( Z^+ -c l - I a 1 )co84^ . 

 L 1 \ A A / \ * / 



The constant term merely gives the relation between a and C '; equate 

 the coefficients of the periodic terms and we get, after dividing throughout 



by a 2 , 



' J 



