886 



„ ^^itü! . 10' = ;i (,«). 10» = — 112732 



+ 2064 cos 2x0 l (^^) 



— 23 eos 4 w. 



4. The first system of differential equations of Chapter II ^ 3 

 of the "Investigations" contains the equations for the determination 

 of the four variables q, a, 0, il, supposing Titan's eccentricity 

 to be zero. The remaining part of Chapter II contains the 

 determination of those terms of the variables (>, a, 0, which 



V M 



are of order zero and one vt^ith respect to IX 7>. To com- 

 plete the computations we shall examine the last equation of the 

 system mentioned, viz. the equation for 12. This equation: 



di d(X 



can be written thus: 



dil m' df 



(20) 



— (21) 



dt a' do ' 



As Q, (J and ^ are known functions of t, the solution of this 

 equation is reduced to a quadrature. 



The right member of the equation is an even function of ; sub- 

 stituting the series from Chapter II ^ 3 of the "Investigations" for 

 Q, <7, ^^, this right member gets the form : 



fx'^ls <P,,iiP; (22) 



_ ,,=0 



here (j. = \/ — and <P., is an even periodic function of t. 

 y M 



Denoting the constant term of the goniometric development of ^^ 

 with respect to t by <P^„ we have : 



-— = ix' 2 cp, IIP + ix' 2 i<P,, - 0,,) fii' , (23) 



dt p=o p=0 



and thus: 



Si = constant + /*' t :s ^piiP -^ -^ iap ((*Pp — ~^p)dr . (24) 



B=0 *' u=zo kJ 



Developing the divisor v according to the series (Chapter II ^ 3) 



ƒ)= OB 



V =1 2 Vp (iP , 



p=:l 



£2 gets equal to the expression : 



