204 Mr. C. G. Darwin on 



and 1 n a 



^2= sec£ + -, 



(-»" 



r 



when 



ce 



x' e _ pc sin ft 



a 



1 H — cos ft 

 r 



and 



r cos 2 £ -i 



v 2 = r 2 + r 202 = c 2 J 1-- - 2 L 



L (l+^cos/3) J 

 Bv elimination of the time we have 



3g; + r ) i ^^ =c { Sln 2 ^ + 2-,cos/3+^cos 2 ^j. 

 Take jt?/r = w ; then 



a , n „ a 2 



w' 2 -f w 2 = 1 -f 2 — cot /3 cosec /3 iv -\ — h cot 2 ft w 2 . 

 p p* 



The orbit is given by 



■/a die 

 4t7= — — -g , 



. /| 1 + 2-cotyS cosec /3w + (^ 2 cot 2 /3-l W [ 



and the time by 



c dt =p cosec ft 1 1 + - cos /3 m? ) dO/w 2 . 



3. The character of the orbit depends on the sign o£ 



a 2 



- cot 2 /3— 1. We distinguish three cases. 

 p z 



I. p>a cot ft. 



The solution is best found by the use of two subsidiary 

 angles. 



Take cot a = - cot ft cosec ft 



P 



and tan 2 ^= tan 2 /jl— sin 2 ft, 



a 2 



so that 1 — -g cot 2 /3 = cot 2 /a tan 2 ^. 



P 



When p is large /jl=%= , and as it decreases they 



