382 
PROFESSOR H. S. HERE SHAW OH THE THEORY OF 
which the area is wrapped. This is, of course, not so convenient, and its possibility 
is chiefly interesting from a theoretical point of view. 
The satisfactory working of the first model proved that the principle of action was 
practicable, and some of the most important results were at once evident; one of 
these being the formation of a chain of such mechanisms in which the loss from 
friction would be inappreciable ; another being the application of a rapidly moving 
screw with clock-work for such instruments as speed indicators. Moreover, the 
compound arrangement might be of very compact form. There was one objection 
which, though of no importance in most applications for purposes of integration, was 
a serious one for certain applications of the converse process.' This was the fact that 
the movable centres, notwithstanding the great range of velocity ratio, could never 
take such a position as to give the limiting values in either direction. This is shown 
at once by fig. 12, where it is seen that to do this, that is, for a to become 0° or 90° 
the movable centres would have to come into contact with the other rollers A A' or 
B B'. It must not be overlooked that although the roller centres are nominally in 
contact at a point, yet that really the sphere, in turning, twists upon the movable 
centres at its equator, while motion of the movable frame causes the supporting rollers 
to twist upon the sphere at its poles. The result is that a smooth, hard, and conse¬ 
quently expensive sphere is required, which in the integrating machine shown in 
figs. 11 and 12 is made of ivory. It should be noted that even if slight wear takes 
place at the centres it is distributed over the whole spherical surface, for directly the 
frame moves round, the former centre becomes a point which, by a sort of precessional 
action, is not likely to again become the centre, at any rate for a considerable period 
of time. 
It was in endeavouring to overcome these difficulties, and also account for the 
satisfactory action of the rolling centres, that the author discovered that the fore¬ 
going arrangement of movable centres was only a special case of a far more general 
principle, which, like it, might, with suitable mechanism, be applied to many bodies 
but which, in the case of the sphere, leads to very practical results. 
Let fig. 17 be the perspective view of a sphere. Suppose A C A' C' to be the great 
circle formed by the intersection of a horizontal plane with the spherical surface. 
Let D C' D' C be the great circle formed by a vertical plane. Then these two planes 
always intersect in a diameter C C', which may have any direction in the horizontal 
plane. Suppose a series of rollers, whose planes of rotation are all perpendicular to 
the horizontal plane, and which are in contact with the great circle A C A' C', sup¬ 
pose also a second set of rollers, whose planes are perpendicular to the vertical plane 
D C D' C', and which are in contact with the circle DCD' C'. Then, by a well- 
known principle of mechanism, rolling contact can only take place between the 
spherical surface and the former set of rollers, when the axis of rotation of the sphere 
is in the horizontal plane, for only then will all their axes intersect the axis of rotation 
of the sphere. Similarly the rollers round the vertical great circle can only roll on the 
