206 



Cincinnati Society of Natural History. 



where » = Gr-—— = 



l-j-e 



2 e 



Since F (E) = ^^ sin^(E— M)+ ^ siii^(E— M)-|- ^sm'(E— M)4- .... 



and F (E') may l^e similarly represented, it is evident by reference to 

 (5) that the last two terms of equations (9) and (10) are of the third 

 order, and have opposite signs. If we neglect these terms, the equations 

 will still be approximately true, and we may illustrate their application 

 b}^ the example given in the Theoria 3Iotus, Book I., Section 10. With 



-^t_ where tan u) = e, and de- 

 1 — e 



Prof. Grunert, we assume tan (45°-|-w) 



rive the following values of our constants ; log tan (45°-j-"')= 

 0-2175146; log (?=log e (1+tan (45°-|-«)) = 9-8129912; log q cosec. 

 \"= 5-1274163. 



log tan ^ M 

 log tan (E'— |M) 

 log cos (E'— pi) 

 log sin ^ M 

 log approx. (E — M)" 

 approx. (E— M) 

 true (E— M) 

 Below is a table giving with the arguments f 

 sum of the terms (E— E')-f-2 F (E'). 



E— E'-f-2 F (E'). 



164° 52' 13"-83 



9-4319633n 



9-6494779^ 



9-9605819/1 



9-4166429 



4-5046411/1 

 —8° 52' 42"-53 

 —8 52 12-14 



arc sin e and M, the 



When M exceeds 180°, the signs of the functions in this table must 

 be changed. 



