SECT. 3.] PLACES IN ORBIT. 13V 



the symbol m or M, with the same meaning that formulas 11, 11*, article 88 give 

 it ; and finally, for the sake of brevity, we write 



c c --- 4 log c 



from which result the equations 

 [13] = (/ 



[13*] M= -( 



which involve only one unknown quantity, 2, since Z is evidently a function of s 

 expressed by the following formula, 



_ (1 + 2 W (* + **)- tog 



100. 



In solving the equation 13 or 13*, we will first consider, by itself, that case in 

 which the value of e is not great, so that Z can be expressed by a series proceed 

 ing according to the powers of z and converging rapidly. Now we have 



3 K 



and so the numerator of Z is f z T -J- | z . . . ; 



3 5 



and the denominator, 2 z* -j- 3 z . . . , 

 whence, 



z=| 1.... 



In order to discover the law of progression, we differentiate the equation 



2(0 + ) l Z=(l + 2r)v/(-ar + ^)-log( 



whence results, all the reductions being properly made, 



18 



