8 



Schubert' s investigation of Ke/pler^3 Problem 



V 



again obtain the same form. Wherefore the third series will be 



I * 



the general form of the {n — l)th differential of all four series ; and^ 



since at each differentiation every term is to be multiplied by the 



coefficient of its angle ai» we shall obtain, in all four series, the same 

 equation, viz. 



which series terminates^ vrhen it arrives at ^ or S^. 



§ 9. Putting successively 



n—fif w=3j etc. and suhstitut 



» 



after having divided it by S''"', in the eciuation (B) 



(§ 7)> we obtain 



(O) 



e=f^-^€ Sill ,«.-f- -— Sill 2/1. + 



e 



+ 



2 



t n /^Nj (7i-2)"-i shi (w-2) A* +N3 (n-4)"-^ sin (ti-4) 



± Nr (ii—2ry~^ sill (71— 2r} ;«, 



which the upper or lower sign, prefixed to N,., is to be used, 

 wording as the number r is even or odd, and the whole series 



Th 



same 



there 



hen n — 2r becomes equal to S or to 1. 

 be used in the rest of this paper, so that, whenever 



sign ± or t, the upper or lower is to be 



double 



made use of, according as the exponent at the bottom of the next 

 following coefficient N, (or whatever it may be,) is an even or odd 

 number. 



as far 



This very essential remark must be well remembered* 



Suppose, for instance, the development of e is desired 



the twelfth power of e, then the equation (D) will 



give 



-K sin A^+|-8in 2/*+l (3 sin 3 ^— sin ^) + ~ (2sin4 M^mn 2 fc) 



e 



r5«,sin5M— 3*.sin S^M^-Ssin /t)-f. 



« 



e 



%*.3.5 



(3*.sm 6)11—2 « . sin 4 f*+5 . sin 2,u) 



+ 





(7*. sin 7 



5» . sin Sm^^i . sin 3m^5 sin ^) 



^ 



j^ 



