Schuberfs investigation of Kepler^s Pvohlenu B 



+ 04 o3 Vr (^^^'^^" S^ — 3^sin 6^4-2^r,siD 4/^ — T sin 2/*) 



e' 



+ 015 .^2 ^^y CS^^sin Da«— ^^sln7ft+23.5^sin5/lt^2^3^7.sinoAt-}-2.^.siu,«) 



2l^32.5.7 

 ^10 



„ X 



+ 08 4 2 y C^^-S'" 10;^--2^ 9.5.sin 8j«.+3ii.5 sin C^2i2.3.5.sin 4^-f 2.S.5.7,sin 2«) 



-^ «o «0 •/ ^ 



■4- ^^ ,, (lP.sitill>t~3^».sin9At+5.7^Psin7/tt-3.5^^sin5itt-}-g.3»^5.sin3/^-2.3.7.sin.tt) 



.J 1_? (2^3^*'smli?;^-5iisinl0«+92ij] sing 3io.5_ii,sJn6^^29.3.5.nrsin4/<^2.3.11sin*t' 



^ ll. In order to develop sin it in a like manner, we must 

 make use of the product cos i^ . sin> (C) (^7). Therefore, the 



I 



series (7) (§ 8) is to be multiplied by cos 2,«, which being done, 

 each term of tiie two first series, which will have the form cos z/i 

 cos kfi, must be put equal to ^ cos (^ + i) (« + -^ cos (Ic — ij (U, and in- 

 stead of each terra of the third and fourth series, viz. cos i^ sin A>c, 

 we must substitute ^ sin {k + ^ + ^ ^in {k ^ i)/a, according to tlie 

 well known rules of Trigonometry. Hence arises 

 (9) S"* cos ifjL sin"/A 



+sin(/7+i>+sin(«-/),tt-NiSin(iz-2+i)jM^N,sin(72-2-i)^ 



Since these series have exactly the same form as that of (7) 

 (§ 8), they will in like manner give 



■ 



(10) iii» ~--^ !r= (7z-|-t)«-isin(/i-fi),«.4- (7?— 2}"-^sin (n — i)f^ 



which series terminates, when n — 2r is equal to or to 1. In 

 the latter case, the last term becomes 



jhNn-i r(l-K)"-^sin(l+i}^-F(l —iy-' sin (1 — i)^] ; in the former, 



tills term is only = ± X„. i"""\ sin i fi. 



§ i2. If we put successively n=2^ n=B^ etc, and substitute the 

 series (10), divided by S", in the equation (C) (§7)? there arises 



2* 



