136 CEETAIN INDICATING AND RECOEDING APPARATUS. 
B, They are both attached to one movable frame, their planes 
always being perpendicular to each other. 
Then ; 
vel. ol C 00 
but by similar triangles OSQ^ ORP, SQ = OR; 
or' 
“tan 
w = IV tan a . 
These may be called the “ sphere and roller ” arrangements 
or mechanisms. The relation between the two forms of disk 
and roller are now quite clear. The first is derived from the 
sine or cosine form of sphere and roller, the second from the 
tangent or cotangent form. 
For in the first case 
when a = 0, w = 0, 
a = 90‘^, 00 = 00 , 
that is the range of velocity is from 0 to oo, and by changing the 
rollers as before from iv to c>o. 
In the second case 
when a= 0, w = 0, 
a = 45°, 00 = 00 , 
a = 90°, 00 = 00 , 
that is the range is as before, from 0 to oo. 
A rough model, with a wooden sphere 6 inches in diameter, 
which was made, and exhibited very clearly the variations in 
relative angular velocity of the rollers as their relative position 
with regard to the axis of revolution of the sphere changed. 
The side motion of the rollers over the spherical surface ' on 
which they have to roll without slipping is a serious practical 
objection to the use of this arrangement, but it might be 
advantageously used for educational purposes to find approxi- 
