iOO Blake on the Teeth of Cog- Wheels. 



the velocities of their planes at the describing point shall be 

 constantly equal, two curves will be traced on the planes of 

 the circles which will drive each other without friction. 



Let a and 6, Fig. 9, be the centres of the two circles, 

 and p the describing point moving towards h ,• and let the 

 space described in a given time on the plane of h by the 

 motion of p ^^pf; and let the space described in the same 

 time in a direction perpendicular to pb by the motion of the 

 circle b about its centre, beyc. And let the corresponding 

 spaces described in the same time on the plane of a, be de, 

 ep. Then the spaces described by the combined motion of 

 the point and circles will be pc, pd. 



Now, if the time be taken indefinitely small, pfc and 

 ped may be considered as right angled triangles ; and 

 since the motion of the point p in the direction of the radii 

 of the circles is common and simultaneous on both circles, 

 pf and ed are equal. And since the velocities of both cir- 

 cles about their centres are constantly equal at the point /), 

 pe and /f are equal. Therefore the third sides pc, pd oi 

 the two right angled triangles pfc, ped are equal, and the 

 angles cpf, pde are equal. Therefore the angle cpf is the 

 complement of the angle dpe. Wherefore the angle cpf is 

 equal to the angle dpf. Hence the curves pc, pf? are coinci- 

 dent at the point p, and since p is any position of the de- 

 scribing point, the curves are constantly coincident, or in 

 contact in the line of centres. Now sinne ed is equal topf 

 ad is equal to af Therefore, in the progress of the mo- 

 tion or action, the points c and d meet and coincide with 

 each other at/. But it has been shown that these two points 

 are equidistant from the point p. In the same manner it 

 may be shown that any other points which are equidistant 

 from p meet and coincide with each other. Therefore, in 

 any giveh space of time, equal quantities of the two curves 

 pc and pd pass through the point of contact p. Wherefore 

 they roll u'pon each other without silding, and consequent- 

 ly without friction. 



Cor. — By varying the relative velocities of the circles and 

 describing point, an infinite variety of atripsic curves may 

 be produced. 



ScHOL. — By a similar process of reasoning it may be 

 shown, that no curves are atripsic, but such as are described*, 

 or are capable of being described in this manner. 



