SECT. 3.] PLACES IN ORBIT. 151 



Putting, therefore, 



we necessarily have 



but in order to decide the question whether J 7 - -J- is equal to-|-2w or _ 2w, 

 it is necessary to inquire whether y is greater or less than c : but it follows readily 

 from equation 8, article 99, that the former case occurs when 2/ is less than 

 180, and the latter, when 2/ is more than 180. Lastly, we have, from the same 

 article, 



+ ) 2 log (y/(l + mm) + m) 

 2 log (y/(l + ) + ), 



the lower signs belonging to the case of 2/&amp;gt; 180. Now, log (\j(l-}-mm)-\-m) 

 is easily developed into the following series : 



This is readily obtained from 



d 

 There follows, therefore, the formula 



and, likewise, another precisely similar, if mis changed to n. Hence, finally, if we 

 put 





- TrfTT (r + ^ + 9) + etc. 



