Blake on the Teeth of Cog-Wheels. 91 



K) 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 ^/, gh, Fig. \^e curves so constructed as to 

 answer the condition of 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. Let 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. It is evident 

 from Prop. 1, that when the curves are both perpendicular 

 to the line ge, an equal additional force applied to the cir- 

 cle b 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 h to resist it; that is it will move 

 with uniform force. 



Secondli/. 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 e 

 and d are equal. But the velocity of the circle a at J is to 

 its Velocity at e, as ad to ae, and the velocity of the circle b at 

 c is to its^-velocity at e as be to be. But be I be'. '.ad I ae. 

 Therefore the velocities of the two circles at e are to each 

 other as their velocities at c and d. But it has already been 

 shown that their velocities at c and d are equal. Therefore 

 their velocities at e are also equal. Consequently if one be 

 uniform the other is also uniform. 



Definitions. — When a circle moving about its centre 

 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. 



When two curves roll together without sliding and any 

 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 describ- 



