CONTINUOUS CALCULATING MACHINES. 
379 
Suppose the fixed centres C and C x (figs. 6 and 7), each in contact always with the 
same point on the spherical surface to be replaced by rollers having horizontal planes 
of rotation. 
It will be seen that as these rollers offer less resistance than the side friction of the 
other rollers, the sphere will be carried bodily round when the angle a is changed, and 
the resistance will now only be that of the turning of the rollers C and Cj. To make 
the arrangement a practical one, and to hold the sphere in its place, idle rollers must 
be placed respectively opposite to A and B, and fixed to the same frame in order to 
counteract their pressure. Also a supporting roller must be placed underneath the 
sphere, with its piane of rotation always perpendicular to the axis of rotation (C C) of 
the sphere, or what is the same thing, its axis must be always parallel to that of the 
sphere, and consequently carried in the same frame as the centres. 
It would not be convenient to actuate the rollers A' and B', or to employ the screw 
axis, if the frame carrying A' and B' actually moved in position, and there is no reason 
why this should be done. 
There are clearly two distinct frames, and two only to be dealt with, one carrying 
the centres and supporting roller, the other carrying the two rollers A and B, and the 
idle ones opposite to them. The motion of the two frames being purely relative may 
be reversed, and the frame carrying the centres made the movable one. 
Thus the sphere will now rotate about movable centres. 
To put these ideas into practical shape, and to see if the rolling centres would 
answer, a model was made to integrate areas. This is shown in plan, and front 
elevation, in figs. 9 and 10. A sphere of boxwood (G) is held in a frame simply 
formed by bending round suitably shaped sheet brass, which thus caused the rollers to 
grip the sphere. The rollers, A A' B B', were disks of boxwood, and merely had 
common red elastic bands cemented upon their peripheries. The paper on which is 
the area to be integrated is folded round a cylinder (M), and held there by two india- 
rubber bands. The cylinder is turned by means of a milled wheel, N, with one hand, 
and the pointer P, which is connected with the movable frame of centres, is kept on 
the curve with the other. The roller A works in contact with the surface of the 
paper, and communicates its motion by frictional contact of the indiarubber to the 
sphere. The motion of the pointer connected with the axis of B is registered on a 
suitable dial. 
The model not only worked entirely in obedience to the movable centres as far as 
its range of action permitted, and required only the application of an extremely 
small force to change the relative position of the frames (i.e., the velocity ratio of 
A and B), but although only roughly made gave approximately correct results. It 
was found, however, that the w r eight of the sphere was not sufficient to keep it 
accurately in its place, and the more elaborate integrating machine shown in plan and 
elevation (figs. 11 and 12) has been constructed. 
In this latter instrument rollers are placed both above and below the sphere, the 
