PROFESSOR T. A. HEARSON ON THE KINEMATICS OF MACHINES. 31 
describes an involute, so the letter C would, in a fairly graphical manner, represent 
this motion. 
A mechanism consisting of two pulleys mounted in a frame joined by a crossed belt, 
may be represented by OcCO, or oCco. If an open belt is used the corresponding 
formula would be OcCo, a small c being used for wrapping, and a large C for 
unwrapping. 
It is not easy, if possible, to make an exhaustive list of these mechanisms. What 
has been said will serve to indicate the method of the application of the previously 
mentioned principles in considering them. 
It will be convenient to class them separately from the plane mechanisms made up 
of the OUI motions. 
Spherical Mechanisms. 
The next division of machines to be described will be those in which the simple 
motions OU only are employed, but in which the axes, instead of being parallel to one 
another, are so inclined that they all meet in one point. 
Referring to the original mechanism, fig. 1, p. 16, suppose the links to be bent so 
that they lie on or parallel to the surface of a sphere, the axes of the pins will all be 
radii of the sphere, and if produced will meet at the centre, and the centre lines of the 
links will be great circles of the sphere. 
Reuleaux has shown that the previously described mechanisms, or some of them, 
have their counterpart in spherical movements ; but he omitted to notice an important 
exception which forms the foundation for a distinct division of machines, in which 
helical motions are employed. 
There are other points of interest and importance, not observed by Reuleaijx, 
which follow from the application of the geometrical laws which govern the association 
of the OU motions in a spherical mechanism. For such movements Law I. must be 
modified as follows :— 
The sum of the four angles of the spherical quadrilateral varies, having a value 
of 37 r for a maximum, and 2ir for a minimum. 
Law II. will remain as before. 
It will follow that each one of the combinations of the OU motions in plane 
mechanisms will have its counterpart in spherical mechanisms. 
Reuleaux has further pointed out that if a mechanism containing a sliding motion 
is adapted to a sphere, then, instead of a sliding motion along a curved link, exactly 
the same relative motion could be produced by the swinging of one Jink about one of 
the poles of the sphere, of which the curved line of slide is the equator. One of the 
two links joined by the slide would then be required to have a length equal to that 
of a quadrant of a great circle. The other may be equal to it or not equal, being 
greater or iess at pleasure, for either pole may be selected for the axis of swing. 
