Ball — Notes on Applied Mechanics. 
243 
It was on account of this decision, without reference to the number 
of vibrations, that I thought of inquiring whether our old fork, em- 
ployed by Handel and all the great composers of that age, and long 
before it and since, was worthy of confidence with regard to clearness, 
purity, and unity of sound. 
XXYII. — ]N"oTEs ON Applied Mechanics. I. Paeallel Motion. II.- 
The Contact of Cams. By Robeet S* Ball, LL. D., M. E. I. A. 
[Read Monday, December 11, 1871.] 
I. PAEALLEL MOTION. 
Theorem. — A plane figure is moving in a plane according to any 
law. There are always two points in the figure, or rigidly attached to 
it, so that the four consecutive positions of each of these points is in 
the same straight line. 
The motion of the figure can be produced by the rolling of a curve 
rigidly connected therewith upon another curve fixed in the plane. 
We may, in the first instance, replace each of these curves by their 
circles of curvature. Thus we may have the figure attached to the 
circle with centre B and radius R' (Fig. 1) which rolls upon the fixed 
circle within centre A and radius R. 
A point F is turning instantaneously about 0, 
and is thus carried to P'. If 0' be the point of 
contact when P reaches P', and if OA 0' = w, 
it is easy to see that 
PP' = IV. OP. 
R + R' 
R' 
If OP' be parallel to OP, the point originally 
at P will continue on the line PP' during the 
rotation about 0\ as well as during that about 0. 
In this case we have 
00'. sin 0 = R ic sin e=oo. OP. 
R\R' 
Ir: 
or 
OP. 
RR! 
R + R 
/ sin 0. 
Fig. 1. 
Therefore P must lie on a circle OPQ touching both the circles 
of curvature at their point of contact, and having a diameter equal to 
the harmonic mean between the diameters of the circles of curvature. 
We learn from this that whatever be the motion of a plane figure 
in a plane, there is always a circle of points rigidly connected with 
the plane figure, such that three consecutive positions of each point 
