S4 



Schuherfs investigation of Kepler's Problem, 



N=0 and N* *=0, therefore A^^>'=0 ; 



ifi 



5 and n 



3, 



J=J^=i=5 ; N=6 ; N» »=]Nro=+l ; A^'^ 



5 



5 



2M.i.3 



• • • 





ifi 



5 and n 



5y 



J=J 



ifi 



n 



5 i A^'>=+ -- 



5 



3M.2..5 



+ 



1 



£8.3 



/ 



6 and n 



N=0 and N^ i=0, wherefore A^'''=0 j 



2, 



ifi 



6 and n 



4, 



J=J 



1; N=o^ ]vii=;^[ ^^j. Ari;=+ 



1 



2«. 2.3.4 



+ 



1 



29.3 



§ 26. Collecting all the above fractions (§ 25) into one sum, we 



shall observe the whole coefficient of 



+ 



2 



+ -C 



11 2 



25 



9 



71 



27.33.5^29 28.3 2". 3^.5 2^,5 



-_ + 



91 



28.33.5 29.3.5 



142 , 819 



,-7- 



29.33.5 29.33.5 



+ 



67? 



29.3^5 



^ 27. We have yet to find the development of the rad 



z, which bein 



5 given by the equation III, (^ i 

 s development will be obtained by the same 

 we substitute in the equation (6) (^ 6), ^ (y) 



1 



e cos €, 



method as above 



Hence, \L 





cos e, 



y 



Vf which b 



g substituted in the equa 



tion (6), gives 



cos t = U + x, «. sin y __ ^. J-siny.(g)« ^ ^^^^ 



1.2 



m 



hich we must put (§ 7) 



dy 



whence we ^'^i 



(F) 



3(~^, tt=cos /<, f =sin ^ J 



=cos fi-e sin^ ;* 





.ti— 1 



1.2 



d 



^ 



• •■• 



d^'-^.S!!!'' u 



§ 28. In order to find the differential, d 



1.2.3.,.(/i_l)* d,i4«— 2 



d iffer 



ential of the first of the series (7) (§ 8) is to be taken twice, the 



. - tmes or not at all, the third three times, and the 

 fourth once, which produces for each of the four series 



second four 



equation 



the same 



