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PROFESSOR T. A. HEARSON OR THE KINEMATICS OF MACHINES. 
Also all the combinations of the I with OU motions previously detailed have their 
counterpart in spherical mechanisms. 
Reuleaux has compared one of the best known spherical mechanisms, known as 
Hooke’s joint, with its plane counterpart, Oldham’s Coupling, p. 27. In the latter 
there are three links which are supposed to be equal and infinite. In the former 
there are three links, the lengths of which are each equal to a quadrant, the fourth 
being shorter. In Hooke’s joint it is well known that the angular velocity ratio of 
the connected shafts is not constant, whereas in Oldham’s Coupling it is. 
It is interesting to notice that these two facts immediately follow from Law’ I., the 
sum of the four angles of the spherical quadrilateral not being constant. 
In spherical mechanisms the relative lengths of the links may be represented by 
the magnitude of the angles at the centre of the sphere which they subtend. This 
suggests that a spherical movement may be represented and distinguished from a 
plane movement by inserting the value of the angles between the letters O and U, 
which represent the motions.^ Thus for examples :— 
The spherical counterpart of OoOo is fiOaofiOaofi, expressing the fact that 
opposite links are equal to one another. 
Of o0 3 oU, the spherical counterpart is /30a0 3 ao/3U/3 where /3 > a. 
Of o0 3 oI it is ■g7rOa0 2 ao^7rU|-7r. a < ; and so on for all the other plane 
mechanisms previously enumerated, with one exception. 
The movement III has no spherical counterpart capable of movement, as it will be 
a spherical triangle, though there is a spherical counterpart to the movement II11. 
It is interesting to notice that the reason which precluded the existence of the 
combination UIII in plane mechanisms, does not hold in the case of its spherical 
counterpart, for the sum of the four angles of the spherical quadrilateral is capable 
of variation. With these exceptions the combinations not possible in plane mechanisms 
are also impossible in their spherical counterparts. 
An enumeration will show that there are only six different ways of combining the 
OU motions in a spherical mechanism, and out of these we can get only twelve 
different movements by inversion. 
Besides these there are the spherical counterparts of those plane mechanisms 
which contain in their composition other motions than OUI. The most notable of 
them is the mechanism consisting of a pair of bevil wheels mounted in a frame, the 
formula for this would be 0aW/30(a -f- /3). 
* The formulas for plane mechanisms may be made to give information about the length of the links 
by inserting figures between the letters which represent the motions. Thus 02'o8'UImay be considered 
to represent a crank and connecting-rod engine of 4' stroke, the length of the connecting-rod being 
twice the stroke, and the line of stroke passing through the axis of the rotating crank. If the line of 
stroke were to deviate by the amount of, say 1', the fact could be indicated in the formula by 02'o8'UlT, 
one of the infinite links being 1' greater than the other. 
