132 
! 
Hence, as the mass diminishes the velocity increases, but 
the kinetic energy in the direction of motion is constant 
none of the energy of the blow is consumed during the 
rotation of the variable cylinder ; once started it would con- 
tinue of itself. In the rolled-up cylinder, there is an 
amount of potential energy which may be estimated as 
follows : suppose that originally we had a thin lamina rest- 
ing on a flat plane ; now, the amount of work necessary to 
raise a particle of weight w to the height y is wy ; and to 
raise an aggregate of particles the work will be 2wy or 
(/My, where y denotes the vertical height of the centre of 
gravity and M the whole mass. In the rolled-up cylinder 
this is stored up as potential energy ; during the motion it 
assumes the kinetic form, and would of itself be sufficient 
to keep up the motion on a smooth plane. In what precedes 
I have supposed the centre of gravity to lie in the normal 
to the plane drawn through the point of contact of the 
cylinder with the plane. This would not be exactly true, 
on account of the cjdinder not being perfectly circular ; there 
will be an extremely small couple due to gravity tending to 
produce rotation. 
If in equation (2) we suppose the length of the tape to be 
infinite, for the time of motion during any finite length we 
shall have t = -. 
v 
In the above problem I have supposed the external edge 
of the tape to be fixed. We may, however, have the internal 
edge fixed and the external in motion, as in the following 
problem. An indefinitely thin lamina is wound round a 
fixed horizontal cylinder of indefinitely small cross section 
to the external edge of the lamina a weight is fixed to 
determine the motion of this edge. Suppose we take 
moments about the fixed axis. Then the expression 
and equal to its initial value. Hence it would seem that 
