36 Schuhert^s investigation of Kepler s Froblem 



§ 31. Nothing now remains but to ai)ply our second 



direct 



method 



general term of which 



Ac 



m 



(§ 39), a being always =7w+Ss. The problem consists in finding 



ay 



be 



a general expression of A, a and m being given ; which 

 performed, by the method above explained, without the least diflB 



culty. 



the proposed term, gives the following equation 



The general term of z (G) (§ 29), being compared 



T 



K 



2«--^' 1.2...(n— 1) 



.6) 



r)"~^ cos (?i — 2r) ^= Ae" cos m M) 



whence we get «=«, n 



5> 



r=m or r 



a — m 

 3 



9. By substituting 



r=s, N^ is changed into N,, and by substituting n=a, N, be- 



comes 



fl(a — 1) (a— 2) ...• (tt— s+l) 



1. 2. 



3. 



$ 



A, (§ 19), wlience we get the 



coefficient of e" cos m ^, 



A 



T 



A^ . m^-^ 



2«-i.i.2...(a--l) ' 



the upper or the lower sign taking place, according as 



s 



a — m . 



t 



2 



is even or odd. 



% S3. Let, for instance, the coefficient A of c' sin 3/t be sought 



then 



i 



a 



7, ?w=3, s=3, A 



* 



A 





3.7, and A 



".7.5^ 



2''.1.2...6 



2»o.5 



. Let the term e'** cos 10 /« be sought : then, 



a 



m 



10,5 



0,A, 



A 



1, and A 



10 



5' 



2M.2..9 



2».3*.7 



If A^" cos /< be sought, we shall have a=il, m 



iHO^? ^^^., , . , 2.3.7.11 



1,5=5,A,==A 



2i'>.l.2..l0 



''on 



11 



2»^3^52 



. If Ae' cos 5^ 



is required, we shall have a=7W=5, s=0, A,=l, and A 



S5 

 2M.2.3.4 



5^ 



2^ 



^ ; all which exactly agrees with § 30. 



