41 



Also Tr = -£^ and p = r + ^'?^^ :. 2pr = Tr^ + p^ 8 



Again if OZ make an angle ^ with the prime vector, equa- 

 tion 6 becomes 



mn(0-x) = ^ ...9 



Hence to determine the Convolute we have 



From the above it is evident that the convolute is similar 

 to and half the dimensions of the inverse pedal of the in- 

 volute of any curve. It has also been noticed by Rev. J. 

 T. Ward, Fellow and Tutor of St. John's College, Cambridge, 

 that the radius of curvature at any point of a convolute, 

 whether unipolar or not, is independent of the curvatures 

 of the curve or curves from which it is derived. 



From equation 10 it is easily proved that if any curve 

 have an involute of the form 



then its unipolar convolute is a curve of the same class and 

 of degree n where 



_ '^ 

 ^~2m + l' 



Also that if the involute of any curve be of the form 



/{p,r)^0 

 where/ is a homogeneous function, then the unipolar con- 

 volute is a similar curve of half the linear dimensions. 



The theory of convolutes leads not only to a geometrical 

 representation of the curves which will produce a given 

 caustic by reflexion, but applies equally to the case of 

 refracted rays. But the details which are based upon the 

 construction of Cartesian Ovals, would become too cumber- 

 some for practical application save for a few most simple 

 values of the refractive index. 



