378 
PROFESSOR H. S. HELE SHAW OH THE THEORY OF 
also 
therefore 
OP=a constant =h 
PF =h- 
®i 
So that P F is a direct measure of the ratio, as is the radial distance of the roller in 
the simple form of disk and roller. Moreover if s= distance turned through by a 
point in the circumference of B in any time 
p Jp 
s=b\ dz=-\ ydx 
Jo o 
where b is the radius of roller B. 
In fig. 7 let P' be again the fixed point, but now take P' F' perpendicular to the axis 
of rotation of the sphere instead of as previously to the plane of roller B, and fix it 
rigidly in this position. 
Employ at F' a sliding swivel instead of a cross as before, through which O' F' freely 
passes, so that as the angle a changes in any way 
PF 
O'P' 
= tan a 
or as before from previous reasoning 
P'F' , u> dz y 
OT 7- tan a 
or 
P'F '=k'~=y 
dx J 
or 
p' j) p' 
s'=6 dz=-\ ydx 
Jo k j 0 
The sphere and roller mechanism might, therefore, be at once employed to replace 
the disk and roller, but with the following important difference. The sine form has 
still the limited range of the corresponding form of disk and roller ; but the tangent 
form, while having unlimited range, has not now the inconvenient relation between 
the variables of the corresponding form of disk and roller, but is as simple to graduate 
and read as the other. 
For practical purposes the graduation along the fixed perpendicular bar of the 
tangent form will be shown to be much more convenient to read than those along the 
oscillating or swinging bar of the sine form, and therefore would probably be generally 
employed. For the present in what follows that form will be the only one treated of. 
Although one of the practical objections, viz., that of grinding, has been obviated, 
there is still left that of side slipping of the roller on the sphere when change of 
velocity ratio is required to take place. This may be obviated in the following manner. 
