Ball — Notes on Applied Mechanics. 
245 
The motion of a cam turning about a centre B (Fig. 4) relatively • 
to the cam turning about the 
centre A consists of an angular 
velocity 0 about the point iV, and 
a velocity of translation V in the 
direction of the arrow marked V; 
these movements compounded with 
the angular velocity w of A must 
give the actual angular velocity iv' 
of the cam about B. 
The relative angular velocity 0 
about iV, and the velocity of trans- 
lation V, compound into a single 
angular velocity about the point X : Fig. 4. 
where XJSf is the normal to the surfaces of contact the angular 
velocity w about A and 9 about X must compound into uo' about B, 
therefore X must lie on the line AB, and by the laws for the com- 
position of rotations about parallel axes which are identical with the 
laws for the composition of parallel forces, we have 
6 = to -If tjo' 
w :iv' :: BX: AX. 
V=X]V0=XJV{iv + w'). - 
We thus deduce the following theorems, the first three of which 
are well known, though the usual proofs do not appear so simple as 
those now given. 
The angular velocities of two cams are in the inverse ratio of the 
segments in which the normal at the point of contact divides the line 
joining the centres. 
For the motion to consist entirely of rolling, the point of contact 
must lie in the line joining the centres. 
The velocity parallel to the tangent of the motion with which one 
cam slides upon the other, is found by multiplying the sum of the 
angular velocities by the intercept on the normal between the point 
of contact and the line joining the centres. 
The motion of each cam relative to the other consists of a rotation 
about the point in which the normal cuts the line of centres. 
