SOLUTIONS OF A SMITH’S PRIZE PAPER FOR 1871. 
[From the Messenger of Mathematics, vol. i. (1872), pp. 37—47, 71—77, 89—95.] 
1. A point moves in a plane with a given velocity, and also with a given velocity 
about a fixed point in the plane: show that the locus is either a circle passing through 
the fixed point, or else a circle having the fixed point for its centre; and explain the 
relation between the two solutions. 
We have in general 
and in the present question, taking the fixed point as the origin, and measuring 9 
from any fixed, line through this point, 
d9 
~T~ — (O, 
dt 
+ r 2 eo 2 , 
where V, w are given constants. Hence 
or, writing V = aw, 
therefore 
or 
d9 = 
dr 
f (a? — r r ) ’ 
6 + ¡3 = sin -1 — 
a 
, (/3 the constant of integration),