54 
Substitute this value in (1), then 
v = \] P - ' Va 2 - x 2 
V w v 
= \Zf- p Q (2) 
The velocity of ( w ) is, therefore, greatest at 0, and zero at 
A and B, and will continue to oscillate between A and B so 
long as the force continues to act. 
It will be observed that the motion of Q is compounded 
of the vertical motion of P and the horizontal motion of the 
line PQ, then, by the composition of velocities 
The velocity at Q = v. ^ (3) 
Equation (3) shows that Q moves uniformly. 
If (t) represents the periodic time, or the time of a com- 
plete oscillation, then, 
a\/-.t = 2arr 
v w 
_ / W 
or, t — \ / — 
* v vg 
which is independent of the amplitude of oscillation. 
The time of the body describing AP = the time of describing 
the circular arc A-Q = / y/^.Sin 
If the body when at A is slightly struck with a force at 
right angles to A B it will then describe an ellipse whose 
centre is 0 — this is Newton’s case of the problem. 
With the above slight alteration in the diagram, the 
reason for which is given above, the solution of this interest- 
ing problem by Professor Osborne Reynolds is simple, 
accurate, and easy to comprehend by means of elementary 
mathematics. 
On this ground it will no doubt recommend itself to the 
favourable consideration of any writer on Elementary 
Dynamics. 
Thomas Alcock, M.D., exhibited and explained a series 
of lithographs of his beautiful drawings illustrating the 
development of the Common Frog. 
