SECT. 3.] PLACES IN ORBIT. 117 



and the denominator, 



SsinMy 12 sin 5 4^ + 3 sin 7 i^ + etc. 

 Whence X obtains the form 



But in order to obtain the law of progression of the coefficients, let us differen 

 tiate the equation 



X sin 3 ^ = 2 g sin 2 g y 

 whence results 



A TT 



3 Xcosff sm z ff -}- sin 8 g =2 2 cos 2^ = 4 sin 2 ^ ; 



y 



putting, moreover, 



sin 2 4 g = x, 

 We have 



i in 

 whence is deduced 



AX 8 GXcosff 4 3^(1 2x) 



dx &in*ff 2x(l x) 



and next, 



If, therefore, we put 



X = |( !-{- x + /? x x + y 3? + &amp;lt;J a; 4 + etc.) 

 we obtain the equation 



f (az-f (2/3 a)xz + (3f 2/3)^+(4(J 3y)^+ 

 = (8 4a) + (8 4/5)*3r + (8|8 4/)^ + (8y 4d)* 4 + etc. 

 which should be identical. Hence we get 



=*,/* = f , f =/ ,* = Hr etc., 

 in which the law of progression is obvious. We have, therefore, 



v | 4.6 , 4.6.8 , 4.6.8.10 , 4. 6.8.10. 12 _, 



= * + 3T5 a? + 875^**+ 3.5.7.9 ^+ 3.5.7.9.1! ^+ etC 



This series may be transformed into the following continuous fraction : 



