We get therefore the equation 



rdd = ldcj 2 



Now ; = £ + ^72ir;^2 3 



therefore # = ?^. + J^frfl _ ,^1 



du) du)^ \/r^-7r^{ dto du)] 



but from equation 1 



dl _d(T _ dr (ZV _dr 



d(t) d(t> dw d(t)^ du} 



hence from 3 and 4 



f _^\ /-^i i ^ ^^'^ 



V dwj ^ "^ ~ ^dix) ^doj 



••• |{ .W^^^ } "'{ J + v'^:^^'} 5 



also d-io = cos-^'^ 6 



r 



Hence if the equation to the curve (V) be ir=f{w), we can 



from equations 5 and 6 completely determine the equation 



to the convolute. It is evident that equation 5 can be put 



in the form 



r+ J^^^^=JiT^w 7 



The equation of the Unipolar Convolute also admits of a 

 simple solution whenever that of the Involute is known. 



For in fig. 3 if Q, Q' be the images of O with regard to 

 the tangents to the Convolute at P, P' and if OZ the perpen- 

 dicular on QQ' be equal to 'p, then the locus of Q is an 

 involute of the curve V, 



