17 
velocity at any point can be resolved into two, perpendicular to 
the radius vector and axis major respectively; and both constant. 
Call these v, and v, respectively. 
Now suppose the body to move from P to Q; when it arrives 
at @ it has velocities equal and parallel to those which it had 
at P, together with the velocity generated by the central force 
during the motion from P to @. Call this last velocity wv, and 
suppose its direction to make an angle with SQ. Let 
PSQ= ¢. 
Thus at @ we have three component velocities in assigned 
directions; and these must be equivalent to v, and v, perpen- 
dicular to SQ and the major axis respectively. Hence 
v, sind —ucosp=0, 
v, cosp+ usin p=», ; 
sin d sin a 45% 
therefore v, (cos b+ Wr cog Ae yal 43 
therefore cos = cos (P—) ; 
therefore ab = $0. 
This result is exact; it shews that as the body moves from 
P to Q the effect of the central force is to generate a resultant 
velocity the direction of which bisects the angle PSQ. 
Let f denote the acceleration at P, then when the time, #, of 
motion is made small enough we have 
fi=u, 
ne 
aht a , where r is put for SP; 
therefore jr sin b = hu 
_ ¥,sin 
~ cosa ’ 
v 
or ultimately Ss. 
