224 
Proceedings of the Royal Society 
For, if AP be the cycloid described by the point P of the circle 
SP rolling uniformly on the line AS, the 
velocity at P is proportional to SP, and 
the direction of motion is perpendicular 
to SP. Hence the hodograph (turned 
through a right angle in its own plane) 
may be represented by the circle SP, 
described with uniform angular velocity 
about the point S. That the motion is due to constant accelera- 
tion perpendicular to AS is obvious from the fact that, if Pp be 
drawn perpendicular to AS, SP 2 cc Pp. 
5. If the orbit be central, and be a circle described about a point 
in its circumference, the hodograph is a parabola described about 
the focus with angular velocity proportional to the radius vector. 
For, if S be the centre of force, P the 
point in its circular orbit, p the corre- 
sponding point of the hodograph : qp , the 
tangent to the hodograph at p , must be 
parallel to SP ; and, therefore, if SQ q be 
the tangent at S, the triangle pSq (being 
similar to PSQ) is isosceles. Thus the 
locus of p is a parabola. Also the angular velocity of Sp, being the 
same as that of PQ, is double that of SP, and is, therefore, in- 
versely as SP 2 . But the length of Sp is inversely as the perpen- 
dicular from S upon PQ, i.e ., inversely as SP 2 . 
6. A point describes a logarithmic spiral with uniform angular 
velocity about the pole — find the acceleration. 
Since the angular velocity of SP 
and the inclination of this line to 
the tangent are each constant, the 
linear velocity of P is as SP. Take 
a length PT, equal to e SP, to repre- 
sent it. Then the hodograph, the 
locus of p, where Sp is parallel, and 
equal, to PT, is evidently another logarithmic spiral similar to 
the former, and described with the same uniform angular velocity. 
Hence pt , the acceleration required, is equal to e Sp, and makes 
with Sp an angle equal to SPT. Hence, if P u be drawn parallel 
