I 



1* 



Schuberfs investigation of Kepler^s Problem 



•i 



, + .TT^ (^ l^'Sin 11,M.— 22.36.sin 9jM,-(-2.3.73.sin T 1.0—9.* .5^ .^m 5jtt+S'.3in S.m.) 

 sin 8£=sm 8iw4-4c(sm Oft— sin r|ic) + 2ea.(5 sin lOjtt— 2».sin 8ft+3 sin 6^*) 



e 



4-:^ ll'.sin llitc— S5.sin9i»4-3.r2.sinr 



a.'' 



.sill 5jM.) 



/ 



2 



+ -e* (2.3^sin 12^6— 5».sln 10itt+2*.3.sin 8;et— 3^sin 6/M,+2sin 4^^); 



^'^ 



9 9 



sin 9.=sin 9/*+ -e (sin 10^^— sin 8/*)+ 53 e^ (u gj^ ll;«— 2.9.sia9^+7.sin T/.*) 



9 



4. _ e* (a^s.sin 12A.^52.sin lOf^+S^.sin 8,«— 3.sin 6^); 



sm 10^=sin 10^+5<sin 1 V~sin 9Ac)+5t2(3 sin 12^0^5 sin 10^+2 sin 8/.*)j' 



11 

 fiin lh=sln 11^+ - ^ (sin 12^— sin 10/tc); sin 13<=sin ISitt. 



I 



I 16. These equations contain every thing necessary for 

 veloping x, as far as the 12th power of e : for, by multiplj 



d 



these products 



ing each 

 % 15 by the correspondent power of a in § 14, and addiii- 



e 



> 



we shall get the required series 



All this will appear evident by an example. 



Let the Coefficient of sin /* be sought : then, whatev 

 tiplied by sin «, in e, in %k sin e, in 4- a» sin 9.. ^\r 



may be united 



into one sum, which sum will be the required Coefficient of 



H 



from 



e 



/ 



8 ^S«.3 



e 



3 



e 



5e8 



7e 



8 



from 2a sin f, e{\+~ + — ^—-^ f. 



4 ' 8 



^2 



a^ 



£10.3^ 







11 



2»7.33.i)2 ' 



s 



7«^ 



.7-3(1+ -r-+:r+-^+ ^^ 



£3 



4 



8 





Si-.^i 



£6 



2»».32 



e 



11 



2 



£»^3^5^^+?) 2»^s3.5, » 



5e* 



from^A' sin 2*, ~- ^1(1 +!!+__ , 



7e« SW 



+ —-) 



gs 



2' 



-> 



:«i 



