Blake on the Teeth of Cog- Wheels. . 97 



■ ^ Cor. — Since {prop. 4. Cor. 3.) the point of action and 

 describing point constantly coincide, the point of action or 

 contact of the acting curves does not constantly remain in 

 the line of centres. 



Prop. 9. — ^^^o Isosagistic curves act upon each other 

 without friction. 



Let gi^ gh Fig. 6. be two isosagistic curves in action at 

 the pointy and attached to the circles a and b in contact at e. 

 Now (by prop. 2. Cor.) during the action of the isosagistic 

 curves the velocities of the circles at e are equal. Conse- 

 quently the circles may be considered as rolling together at 

 the point e. Then (by prop. 3.) the point a considered as a 

 describing point traces a curve ad on the extended plane of 

 the circle b which is constantly perpendicular to the line ae 

 —that is, to the line of centres. Now let it be granted that 

 in the mean time the curve gi rolls on the curve gh without 

 sliding or friction. Then the point a considered as a des- 

 cribing point traces on the extended plane of the circle b a 

 curve which is constantly perpendicular to the line ag. But 

 the line ag does not constantly coincide with the line ae, for 

 (prop. 8. Cor.) the pointy is not constantly in the line ab. 

 Therefore the curve ad is constantly perpendicular, at the 

 same point of it, to two different lines which do not constantly 

 coincide. But thig is impossible. Therefore the curves gi, 

 ^^ do not roll upon each other without friction. The same 

 may be proved of any other fellow isosagistic curves. 



Epicycloids. 



Scholium. — When the generating curve is a circle and the 

 describing point is in the circumference, the curve genera- 

 ted on one of the fellow circles is an exterior epicycloid and 

 that on the other is an interior epicycloid. When the diame- 

 ter of the generating circle is half that of the circle on which 

 the interior epicycloid is described, the interior epicy- 

 cloid is a straight line tending to the centre of the circle; 

 and the forms traced on the planes of the circles are those 

 which are recommended in the practical treatises, for fellow 

 teeth which are designed to act either wholly before or 

 wholly after the line of centres. When the interior epicy- 

 cloid is the driving curve the action is before the line of cen- 



Vob.VlI.— No. I. 13 



