EVOLUTE AND INVOLUTE. 349 



Let ABC be the given curve, and A'B'C' the evolute, 



BB' being the radius of curvature of the given curve at B, 

 and GO' at C. Then if A be the fixed point on the evolute 

 from which the arc is measured, we have 



AS a + c'c= 



' = BB'-CC' 



therefore 



Hence, if a flexible string of length I be fastened at A' and 

 placed in contact with the evolute A'B'C', then, as the string 

 is unwound from the evolute, the free end of it will describe 

 the involute CBA. From this property the names evolute 

 and involute are obtained. 



In . the figure as s' increases p diminishes and we have 

 s' + p = a constant ; if ' be measured in the direction from G' 

 towards A', then s and p increase together and we have 

 s p = a constant. 



It will be observed that a curve has only one evolute ; but 

 a curve has an infinite number of involutes, for in the equa- 

 tion s + p some constant, the constant may have any value 

 we please. 



332. The following polar formulae for determining the 

 evolute of a curve are sometimes useful. 



