110 DIFFERENTIAL AND INTEGRAL CALCULUS. 



centres of curvature is called the evolute, namely, that the 

 original curve may be generated by a point in an inex- 

 tensible string which is kept tight and wrapped on, or 

 unwrapped from the evolute. 



For suppose 00' (fig. 32) is the curve locus of the 

 centre of curvature, the centre of curvature at P, 0' at 

 P', P0 = pj P'0' = p'j cr the arc BO measured from 

 some fixed point of the evolute, then since, in this figure, 

 a increases as p decreases, we must take the negative sign, 

 and therefore 



therefore arc0 + OP= 



or 0P-0'P=arc00', 



or if P0 be imagined a tight inextensible string, this 



string will just wrap on the evolute as P moves along 



the arcPP'. If, on the other hand, cr be measured in 



the opposite direction = arc AO' 0, then cr and p increase 



together and the positive sign must be taken, hence 



hence PO - arc-4 = P0' - arc -4 0', 



or 



as before, of course. All this may easily be proved geo- 

 metrically by considering a curve as the limit of a polygon, 

 and considering the curve traced out by a point of a string 

 unwrapped from the polygon. 



The original curve is called an involute of the locus 

 of the centres of curvature. 



It is manifest that every curve will have a definite 

 evolute, but an infinite number of involutes forming a 

 system of parallel curves, such that P being any point 

 of one of the curves, the others may be traced from it 

 by always marking off the same fixed length along the 

 normal at P. 



Several other formulas for the radius of curvature may 

 be deduced from (1), (2), (3). 



