Blake on the Teeth of Cog-Wheels. 91 
to the line which joins, their point of action to the point at 
which the circles are in contact, and if the driving circle be 
moved with a uniform force and velocity, the other will 
move with uniform force and velocity. 
First. Let gi, gh, Fig. 1. be curves so constructed as to 
answer the condition ot the proposition. By the preceding 
proposition, when the circles are acted upon by forces which 
are equal in opposite directions at the point e they are in equi- 
librio. t an additional force be exerted upon the circle a. 
Then the equilibrium will be destroyed, and the curve gi 
moving toward e will drive the curve gh. before it. Itis evident 
om Prop. 1, that when the curves are both perpendicular 
to the line ge, an equal additional force applied to the cir- 
cle 6 will restore the equilibrium; but by the Hypothesis the 
curves are constantly perpendicular to the line ge. There- 
fore if the additional force exerted upon the circle a be uni- 
form and constant, it will constantly require an equal addi- 
tional force upon the circle 4 to resist it; that is it will move 
with uniform force. : 
Secondly. Since the corresponding points g in both curves 
are in contact with each other, their velocities at any given 
instant of time in the direction ge are equal. But the velo- 
Cities at d and c in the direction ge are respectively equal to 
the velocities at g in the same direction. ‘Therefore while 
one curve drives the other the velocities of the circles at ¢ 
and d are equal. But the velocity of the circle a at d is to 
its velocity at e, as ad to ae, and the velocity of the circle 6 at 
eis to its velocity ate as be to be. But be: be: iad ? ae. 
Therefore the velocities of the two circles at ¢ are to each 
other as their velocities atc andd. But it has already been 
shown that their velocities at c and d are equal. herefore 
their velocities at e are also equal. Consequently if one be 
uniform the other is also uniform. 
Derinirions.—When a circle moving about its ¢ 
with a uniform force and velocity drives another circle by 
the acting curves, with uniform force and velocity, we shall 
call the acting curves Isosagistic; and with reference to 
each other, fellow isosagistic curves. ; 
point in the plane of one traces on the plane of the other, 
another curve, the curve thus traced is said to be GENERA- 
TED; the point by which it is traced is called the pesumis- 
