The Inverse Method of Central Forces. 105 
PROP. V. 
‘If a body revolve round a centre, and be acted 
upon by a force tending to that centre which varies 
as the nth power of the distance inversely, and whose 
quantity at an apse, the distance of which apse be- 
ing 7, is= 1: likewise; if another body, besides 
being subject to this force, be acted upon by an ad- 
ditional one, which varies inversely as the gth power 
of the distance, its value being ¢ at the same distance 
r; it is required to investigate a general expression 
‘for the ratio of the angular velocities of the two 
bodies when at the same distance from the common 
centre of force. 
Let the required ratio be that of Fi G. Put y 
== common distance, # = perpendicular upon the 
tangent to the orbit described by the single’ force, 
and » = perpendicular upon the tangent to the 
other orbit, drawn from the common centre. 
It is evident by Prop. 1st, that F* } G* +t 
. ‘ z t 
pe 58 + p ; J f 4+ p : + ce ee 
Tay oe en ; ++ c P RY 
sag jb aNeiede tow 
™m x r , a 
But (by Prop. ist and 3d) ¢* = = : 
VOL. Ve re) 
