38 
“On an Elementary Solution of the Dynamical Problem 
of Isochronous Vibration,” by Professor Osborne Reynolds, 
M.A., F.R.S. 
When a heavy body is free to move in one direction, 
subject only to a force which is proportional to the distance 
the body has moved from some neutral position and tends to 
return the body to that position, the body will, if set in 
motion, vibrate about the neutral position in a period 
which is independent of the magnitude of the motion. 
The deduction of this theorem from the laws of motion, 
although well known, is generally accomplished by the solu- 
tion of a differential equation. In some text books this is 
avoided by comparing the law of force on the vibrating 
body with that of a component of the centrifugal force on a 
revolving body; this method involves no mathematical 
difficulties, but it is indirect and hides rather than removes 
the dynamical difficulties. My own experience has shown 
me that the mathematical difficulty or obscurity of these 
methods stand very much in the way of those who are 
commencing the study of practical mechanics, in which 
vibration and oscillation play a part of fundamental import- 
ance. It was with a view of meeting the requirements of 
such students that I sought for a method involving only 
Elementary Mathematics, in which the solution depended 
directly on the principle of the conservation of energy. 
Having succeeded in finding such a method which, although 
it bears a superficial resemblance to the method of the text 
books already mentioned, so far as I am aware, has not 
hitherto been published, it seems that it may be useful to 
publish it. 
The method is to show that the vibrating body will at 
all times be opposite, in a direction perpendicular to its path, 
to a body revolving uniformly in a circle, having a diameter 
equal to the amplitude of oscillation, with a velocity equal 
to the greatest velocity of the vibrating body. 
