306 Note on Successive Involutes to a Circle. 



I conclude with the remark that if we regard the s and r 2 of 

 the successive involutes as rectilinear coordinates to a variable 

 point, the arco-radial equation will represent a peculiar class of 

 unicursal algebraical curves. Thus the first involute will repre- 

 sent a pair (one for each branch of the curve) of coincident right 

 lines, and the general second involute (taking r 2 and s 2 as the 

 coordinates) a pair of coincident semicubical parabolas. 



In making s vary continuously on passing a cusp, the corre- 

 sponding abscissa from increasing must begin to decrease, or vice 

 versa, according to the principles previously noticed. 



Thinking of the recovery of the cusps and apses from the arco- 

 radial equation, I have been led to consider a morphological 

 property of a more general class of unicursal equations, which I 

 think is likely to bear valuable fruit, and may possibly form the 

 subject of another communication. 



Athenaeum Club, 

 September 1868. 



of curvature is everywhere equal to its elongation from the centre ; that it 

 is a trajectory to a central force varying as the inverse cube of the shortest 

 distance from the periphery of the originating circle ; that its arco-radial 

 equation is of only half the number of dimensions of the general involute of 

 the same order ; and that by the simplest form of quadratic transformation 

 (viz. that which leaves unaltered the inclination of the tangent to the radius 

 vector) it changes into the half-pitch circular convolute ; not to add that its 

 polar equation is even simpler than that of the first involute. Certainly, 

 then, as it seems to me, it ought to take permanent rank among the spirals 

 which have a specific name on the geometrical register; and for want of a 

 better, with reference to the place where its properties first came into 

 relief, it might be termed the Norwich spiral. Where it meets the first 

 involute we have 



or 



(<p4-l)(<£ 2 -3)= ; 



so at the real intersections the radius vector is 2a, and the perpendicular 

 on the tangent, viz. ( — — a J , is a, showing that the tangent and radius 

 vector at those points are inclined to each other at an angle of 30°. 



