
1X.—On the Toothing of Un-round Discs which are intended to Roll upon each 
. other. By Epwarp Sane, Esq. 
(Read April 16, 1877.) 
The construction of the toothed wheels used in machinery gives rise to some 
very interesting investigations in the geometry of motion. The general problem 
is so to shape the contours as that they shall remain in contact while the wheels 
_ turn on their centres with uniform angular velocities. 
The inquiry becomes more extensive when the velocities of the wheels are 
to be variable ; as, for example, when we seek to imitate the revolutions of the 
planets round the sun, and for that purpose introduce the equation of the 
centre. 
In these cases the wheels are supposed to turn on fixed centres ; but we 
may still farther extend the scope of our researches by removing the centres 
and subjecting the discs to the single condition that they roll upon each other. 
If two discs A and B touch at the 
point S, and if they so move as that 
the point of contact shifts equally 
along the two boundaries, they are 
said to roll on each other ; that is to 
say,if wemeasure equal distances ST, 
SV along the two boundaries, the 
points T and V will come together 
in the course of the movements. 
This rolling may be effected in 
various ways. One of the discs, as 
A, may be held fast while the other 
is rolled round about it. In this case the motion of B is regulated by its own 
form as well as by the form of A. In order that the motion of each disc may 
depend on its own shape alone, we may imagine the point of contact S to remain 
motionless, while the contours of the two discs slide past it in such a manner 
as to be continually perpendicular to a fixed straight line. In this way when 
VOL. XXVIII. PART I. 3D 
Fig. 1. 
