Schubert^ s investigation of Kepler^ a Prohlem, 



19 



4 



MA 



("0 



+ 



M, 



2«-i.)tt 



, § SI. The remaining part 





1.2, ^K 



2 



[(?i-2r+i)"-' sin (n^^r+iV+(u-.S?^-i)"-^sin (n-^r-f )aj 



can be made to agree with the proposed term e"" sin m (tj by tliree 



different suppositious 



i)n 



%r + i 



m 



y 



2)n — 2r 



I 



m 



9 



S]n 



2i 



t 



m 



? 



whence we get 



l)2r 



» + 2 



m 



^ 



«> 



r 



w 



2 



?w 



? 



3l3r 



?i 



? + m; 



of which, however, no more than two values of r can take 



hatever may be the numb 



F 



qual to niy or less, or greater than m. In the first case, 1} and 



give the same value r = j?the other being r 



n 





m. 



In the 



second case, 3) is impossible, because 2r can never be greater than 

 n [% SO) ; wherefore, there remains only r = — - — and 



r = — r — • In the third case, l)is impossible, for the same 



2 



reason, and we have only r 



n 



Ml 



2 



and r 



71 — i 4- m 

 3 ' 



The first 



case, r 



— andr 



2 



n 



2 



m, is comprehended in the second case, 



where r 



n-\-i — m 

 2 



and r 



n 



m 



2 



J if i becomes c(iual to m. 



Thus, we have only the three following values of r, 



i)r 



n+i — m 

 ~2 



2)r 



n— I — m 



3 



3)r 



the last of which gives n 



2r 



I 



n — i -\- m 



m, whence we get 



Sr 



:\n— 1 



i) 



siQ(?i 



2r 



i)^ 



mY~^ sin ( 



nif^) 



rnf~^ sin »z /t 



m 



n— 1 



l)''~^sinm;a 



+ ni 



n-l 



1 )" sin w fi. 



\ 



