Blake on the Teeth of Cog-Wheels. 99 



pitch circles. (See Fig. 7.) If the generating straight line 

 be considered as a circle with an infinite diameter, the curve 

 described on one circle may be considered as the last of 

 the exterior epicycloids, and that on the other as the last of the 

 interior epicycloids, and their action is strictly epicycloidal. 

 Another mode of action between involutes of circles was 

 discovered by Professor Robison of Edinburgh, and recom- 

 mended to be adopted in the formation of the teeth of 

 wheels. Let a and h be the centres of two circles whose 

 circumferences are at any short distance asunder, and let a 

 line cd, touching them both, move forward and drive them 

 by simple contact. Then any point p of this line traces on 

 the planes of the circles, involutes capable of transmitting a 

 uniform force and velocity. It will be observed that these 

 involutes are not generated on the pitch circles, and there- 

 fore do not come within the limits of the principle of epicy- 

 cloidal action, as given by those who preceded Professor' 

 Robison : for they had only shown that epicycloids were 

 isosagistic when generated on the pitch circles. The Pro- 

 fessor was therefore clearly entitled to the credit of having 

 made a new discovery, though Dr. Brewster and others 

 were disposed to withhold it from him ; for no connection 

 can be traced between these involutes and epicycloids, in 

 general, as it regards their isosagistic properties, except 

 through the medium of the general principle, as given in the 

 preceding propositions. And it is to be observed that these 

 involutes are not excepted out of the general truths al- 

 ready proved with regard to isosagistic curves ; for though 

 they are not generated on the pitch circles, as we have thus 

 far spoken of them, yet like all other isosagistic curves they 

 are susceptible of being generated by a curve rolling on the 

 pitch circle. It is worthy of remark, that the curve by 

 which these involutes mightbe thus generated is the Logis- 

 tic or Logarithmic Spiral. This truth might be easily de- 

 monstrated, were it not going farther than I had intended i(i- 

 fo the minutiae of the subject." 



Atripsic Curves. 



Prop. 1 1. — If a describing point be made to move in a 

 Jine which joins the centres of two circles, and if at the same, 

 time the circles be made to revolve in such a manuer that 



