98 Make on the Teeth of Cog' Wheels. 



tres, and when it is driven the action is behind the same 

 line. 



When the generating curve is made up of the arcs of two 

 circles, one of which is convex and the other concave to- 

 ward the centre of one of tiie base circles, the diameter of 

 each of these arcs being half that of the base circle toward 

 whose centre it is concave, and the describing point being at 

 their point of junction, the forms traced are those which are 

 recommended for fellow teeth which are designed to act 

 both before and behind the line of centres. 



Prop. 10. — In the preceding instances and also in all oth- 

 ers, when the interior epicycloids are generated by circles 

 less than the base on which they roll, the friction between 

 the interior and its corresponding exterior epicycloid is as 

 the DIFFERENCE of their lengths ; and when the generating 

 circles of the interior epicycloids are larger than the base on 

 which they roll, the friction is as the sum of their lengths. 



For in the first case, every part of the shorter curve is ap- 

 plied to a corresponding part of the longer one. Now if the 

 shorter merely rolls on the longer without sliding, it is ap- 

 plied to it only to the extent of its length. Consequently 

 the shorter curve must slide upon the longer to the extent 

 of their difference. In the second case the interior epicy- 

 cloid is on the exterior of its base, and it is evident that each 

 curve slides entirely over the whole of the others^ Conse- 

 quently the friction is as the sum of their lengths. 



CoR.^Hence the friction is the same, whether the 

 action is before the line of centres or behind it. 



Scholium. — The rule for determining the quantity of 

 friction between epicycloids, as exhibited in the preceding 

 proposition, is applicable to all isosagistic curves whatever. 

 The rule may be thus stated : The friction between any 

 two fellow isosagistic curves is as the difference of those 

 parts of them, which are described by a generating curve 

 more concave than the circumference of the base circle 

 toward v/hose centre it is concave, added to the sum of the 

 other parts. 



INVOLUTES. 



Scholium. — When the generating curve is a straight 

 line, the isosagistic curves generated are involutes of the 



