ss 



Schubert^ s investigation of Kepler^s Problem 



^)i 



\ 



% n 



2,n 



% Tfl 



5) i= 5jn 



i^ n = Sy n 



S- 



5; 



Q}i = 6, n = 2p n 



4; 



• 



the two first cases will eive A^'' = -^~ —, and the following 



a^o-M.a.-.n 



four A'^ 



J(±N±Ni»(— 1)") 



2^»-M.2.3 



n 



, because i is srreater than m or 3. 



In the third and fifth cases^ n being an odd number, we have A^'^ 

 «)io-» . o -' J>ut in the fourth and fifth cases, where n is even, 



2^" ".1.2. ... n ' ' 



A^^^ is 



with i 



J 



than 10 



■ lo-n , o -* The coefficient of e^'* sin 2 a* terminates 



6, since any value of i, greater than 6, would make 



0, or both N and N" = 0. For, since n cannot be greater 



i) as we have seen, n must needs be less than % i be- 



i + 2 become nega- 



t 



S as well as n 



ing >- 6 ; but then, n 



tive, whence N=0 and N"=0 (§ 33). The account 

 to the above formulse of A^^-*, stands thus : 



according 



J=J 



1.6.7.8 

 1.2.3.4 



if 2=1 and n 



S.7, N^=No=+l, N=0 



; 



therefore Af'^ — + 



2.r 



2 



+ 



28 



if 2=1 and n 



J=J 



1.5.6 



1.2.3 



5; N^=N, 



n 



3 ; N=]sr, 



+ 1 



A--'^ 



5.2 



2M.2.3 



5 



if 2*=1 and n 



2^3 



1.4 



5.4 



J=Js=|:^=2;Ni=N,=-}-"-:2^+10;N=N,:=-n=.5; A^'J=+ ^^" 



I 



2»4.2,.5 



+^ 



ifi 



2^3 



1 and n 



i 9 



J=Jl=i=l;Ni_.^ 



3 



7Jj^5 7 



o 



.7 





1.2 



Ml 



1 aivl ?7, 





• «* 



9, 



1 



_1 



r^ 



