j 
Preston—A pplication of Parallelogram Law in Kinematics. 471 
as before), it follows that the velocity at P’ is the resultant of that at 
P, compounded with a velocity perpendicular to PP’, and propor- 
tional to it, viz. w. PP’. Remembering the definition of accele- 
ration as rate of change of velocity, we find at once that the 
resultant acceleration is perpendicular to the are, and w times the 
rate at which the arc is being described, thats, ww=w'r. Thus the 
centripetal acceleration in terms of the angular or linear velocity is 
Oo — a — eh 
r 
This is the fundamental proposition of the whole subject, and 
in the following we shall make constant use of it. Expressed in 
words it implies that when a point is describing a curved path the 
accelerations directed towards the centre of curvature is wv or w’r, 
where w is the angular velocity, and r the radius of curvature. It 
may also be expressed as 2*/r, which is the square of the velocity 
multiplied by the curvature, and shows how this centripetal accele- 
ration at right angles to the direction of motion depends on the 
curvature of the path that is on the angular velocity or the changing 
0. 
Fig. 4. 
of the direction of motion. It is for this reason that when the 
axes of reference are a mutually rectangular system the accelera- 
tions parallel to the axes are independent of each other. 
It is otherwise when we use polar coordinates, and to exemplify 
this we shall apply our method to determine the accelerations 
of a moving point, estimated at any instant, parallel to and 
perpendicular to the radius vector. Let PP’ (fig. 4), be an 
SCIEN. PROC. R.D.S., VOL. Iil., PART V. Ze 
