Prof. Sylvester's Astronomical Prolusions. 65 



% i 1 being the angles which p, p' respectively make with the 

 normal ; i. e. 



„/ i-<«+p» \ £ ( 1 ~w +p ) 



Hence, in order to satisfy this identity, we must have 



1-1 



or 



so that 



^(Ipv^-v 





And accordingly the required identity will be completely satisfied 

 if we further make 



- = —, or„=5, 



which implies 



^ = cV, or ^ = 4, 



c, 7 being the distances of the respective centres of force from the 

 centre of the orbit. 



The vis viva consists of two equal parts, ~^ t ^-—, each centre 



contributing, as it were, equally to its production. To find the 

 time, calling u the angle which the orbit swept out subtends at 

 the centre, we have 



/du\ q _ fi 



[diJ -J? 



or 



t= \- — ~~ = — I du{\ + c 2 — 2c cos?/) ; 



op 



and P, the periodic time, will be — (14 c~), or, restoring the 



ft* 



Phil. Mag. S. 4. Vol. 31. No. 206. Jan. 1866. F 



