, ^ 



Schuhert^s investigation ofKepler^s Problem. 

 If, in the second place, sin ii be sought (§4), we shall have 



7 



'^{y)=siniy—sin it, ,j^'= "^ 



d 



i.cos i e, y=a=.f*.y u = 4^ (a) = sin i M>f 



and the equation (6) becomes 



(C) sin it = sill i/* -f- ci . cos tV sin ft -f- 



. (1 . cos I/* .?in*/tt. 



1.2 



.1. 



dAc 



+ 



« • 



+ 



e" 



L » '^••fl«f^ 



.1. 





Therefore we must seek a general expression of the differentials 

 d"~' . sin '>, and d"~^ . cod «^ sin "^ ; for which purpose we must de- 

 velop the quantity sin > or (sin /*)" into a series, proceeding accor- 

 ding to the multiples of the angle fi, 



§ 8. Analytical Trigonometry affords four different series, ac- 

 cording as the number n has the form 4r, 4<7'-f^, 42'+l, or 4<r-f-3, 



VIZ. 



(7) 2 



71—1 



sm"^ 





f +COS 71^ — ^NjCos (n — 2)i«»+lS'3Cos (?i 



cos n/^'+NjCos (n — 2)^^ — N^cos (n — 4)5* + .... 



Nn-2 COS 2.«^i-|^!L? 



2 3 



N n-2 COS 5.^+^Nrt ; 



2)jM.^NgSin(n — i)/^ — *.»-f-Nfi-i sin ft; 



2 



2 



sin «A«^-f N J sin (11 — 2)/^ — N,sin (n — 4) ja-f .... + N n-i sin f*. ; 



N» 



n 

 1 



_ r»(«-.1) ^.„ „(n-l)(n-2)...(T+l) ^ 



1 •^•O.**** -~- 



being the well known coefficients or multiplers of each term of a 



Binomial raised to the power n. 



In order to find the (n — 4)th differential of these series, the 



differential of the first must be taken (4r 



n 



the 



d 



'+!) 



third 



times, because 

 (4:r) times, the 



fourth (4r+2) times, and thlB form of the differentials will be dis- 

 covered, as soon as the differential of the first series is taken three 

 times, the second once, the third four times or not at all, and the 



fourth twice, because after four 



J 



differentiations 



% 



