374 
PROFESSOR H. S. HELE SHAW OH THE THEORY OF 
wheel B, which is likewise driven at a speed which varies with the other variable 
quantity. The wheel B, though not rigidly connected with N, compels it by means 
of its square or polygonal section to rotate with the same angular velocity, while at 
the same time allowing it to slide freely in a longitudinal direction. If, now, the 
wheel A, or what is the same thing, the cylinder, has the same angular motion as the 
wheel B, then the nut N remains at rest. If the motion is not the same, the nut is 
moved by the action of the screw until it is so. The position of the centre of the 
globe, or of the index I on the scale S, indicates at a glance the ratio of the 
increments of the two quantities. 
Just as in the case of the disk and roller, by causing the motion of the wheels A 
and B to be sufficiently rapid, the differential coefficient may be approached with any 
required degree of closeness according to the equation 
T _ (o dz 
yK=—=—, 
a) x dx 
where y is the distance of the centre of the sphere from the centre of the disk. If 
this mechanism were applied to measure velocity, the wheel A would be driven 
directly by the clock, the wheel B by a machine whose velocity is to be measured. 
Another mode of dealing with the resistance of the roller to sliding has been 
suggested by Mr. Vernon Boys,* in connexion with a different mechanism. This 
method consists of a very ingenious device which the inventor terms a “ mechanical 
smoke ring,” but though described by means of a detailed drawing it does not appear 
to have been actually constructed, and indeed it is not easy to see how this could be 
accomplished to ensure a piece of mechanism giving accurate results. 
(2.) It is now necessary to consider the second kind of objection, viz., the limited 
range of measurement of the disk and roller. This does not affect the magnitude of 
the growth of either variable, but it does affect the measurement of their ratio, which 
is given by the radius of the circle on which the roller or sphere is turning on the 
disk. In theory it is only necessary to alter the units and reduce the scale to any 
required extent. This, in the integrating form, merely results in magnifying the errors 
near the zero position, that is near the centre of the disk. In the converse application 
it has a more serious result, which must be briefly considered. 
Bearing in mind that it is the ratio of the rate of change of the two quantities 
which is being now considered, let it be assumed that this with each of them in turn 
becomes relatively very small. 1st. Let this take place w T ith the one which regulates 
the motion of the screw; in which case the roller B, fig. 1, simply moves inwards 
towards the centre of the disk, and will register to the limiting value however small. 
That is in the notation already adopted, since 
&>i 
* ‘ Philosophical Magazine,’ vol. xxii., p. 80. 
