ScliuherVs investigation of Kepler' s Problem, 



fja 



g„_i d"~^.sin"iM, 



d/*" 



n 



cos n fi +Ni(w-S)"-=coa (w-S)<u 



• • 



T N, (w— 2r)''-=^ cos (k 



f> 



J')-", 



in which r must always be taken less than |^. Hence the equation 



(F) becomes 



e 



cos t=COS (I +-^(C0SS;M 



I e 



2 



l)+r.- 



22 1.2 



+ 



2 



1 e"-i r 



'^*l.2...(r6-i)[_ 



«"~^ COS ?l fJL 



N.rn 



4 



(3 COS 3 (A. — 3 cos (Jt) + 

 3)'-*-^cos(n— S)^+ .. 



± 



N,(n 



Sr)"-" cos (?i— 2r) ^ 



§ 29. This vahie, being substituted in the equation 



o 



1 



e cos e, gives 



(G) »=l-e cos ^-^ (cos S^ 



1 _e 



22* l.a 



(3 cos 





3 cos ^) 



• #*• 



I- 



1 



a 



n-l" 



1 





cos ?i /i — Ni {n — S)"-* cos [n — 2) ^ + 





Sr)""^ cos [n — 2r) ^a 



=.f 



^ 30. The development of this formula as far as the twelfth 



power of e^ gives 



zz=zi -ecosjtc 



e2 



(cos 2^-1) 



8 



c5 



(cos 3jC4 - COS A4) " — (cOS4At-COS :2jt4) 



s 



6e^ g« 



(52 .COS 5jtc — 3 '.cos 3^+2 cos/t)^ — ^— 3^ .cos 6jtt— 2^.cos 4^+5 cos 2^) 



3^.3 



2*.^ 



7^7 



10 02 g (^'*-cos r^t — S^.cos 5^+3*. cos 3/*^ — 5cos.«.) 



3^".3^.5 



3 



2^.3^5.7 



(S^.cos 8^4 — 3®,cos6.w-f 2*,7cos 



T 



cos 2itt) 



e 



9 



i^l^5.7 



(3^2,cos9,«.— 7^.cos 7it6+2* .e'^xos 5<^— 2^ 3«.7.co3 3w.+2.7-cos /«,) 



€ 



10 



8 77T"-(^^'^°2 1<V— 2^'^.cos 8^+3^^,cos 6/a— 2**,3.cos4i6t+2,3.7.cos3;it) 



lie 



1 1 



Q.8 .-.^ ,., , (11 ^cos 1 ljtc-3 1 8 .COS 9i«.+5.r».cos rjM.-3.5i<'.co3 5^*4-2.3 ^ ".J.cos Sju,-2.3.r.cos /*) 

 —-——-:—, (28.3».cos l2^-5l^cosl0/6+3*'.U.cos8/c-a^5.11.cO3 6;t+2^3.5.11.cO3 4A<^2.3.11.cos 2^) 



