Prof. Cayley on the Theory of the Evolute. 347 



of the class 3 and order 3, the reduction in the order from 

 3.2, = 6, to 3 being caused by the existence of an inflexion. 

 Comparing with the former case, we see that the effect of the 

 contact of the conic with the absolute is to give rise to an in- 

 flexion of the evolute, and to cause a reduction = 1 in the class, 

 and a reduction =3 in the order. 



I return, now to the general case of a curve 



Using, for greater simplicity, the equation # 2 + y 1 + £ 2 = for 

 the absolute, the equation of the normal is 



n= 



* ,y if 



a,u, a,u, d*u 



=0; 



we may at once find the class of the evolute ; in fact, treating 

 (X, Y, Z) as the coordinates of a given point, the two equations 

 TJ = 0, = determine the values (x, y, z) of the coordinates of 

 a point such that the normal thereof passes through the point 

 (X, Y, Z) ; the number of such points is the number of normals 

 which can be drawn through a given point (X, Y, Z), viz. it is 

 equal to the class of the evolute. The points in question are 

 given as the intersections of the two curves U = 0, 11 = 0, which 

 are respectively curves of the order m, hence the number of in- 

 tersections is = m 2 . It is to be observed, however, that if the 

 curve U = has nodes or cusps, then the curve XI = passes 

 through each node of the curve U = 0, and through each cusp, 

 the two curves having at the cusp a common tangent ; that is, 

 each node reckons for two intersections, and each cusp for three 

 intersections. Hence, if the curve U = has 8 nodes and k 

 cusps, the number of the remaining points of intersection is 

 = m 2 — 28 — 3/c. The class of the evolute is thus = m 2 — 28 — 3/e . 

 The number of inflexions is in general =0. If, however, the 

 given curve touches the absolute, then it has been seen in a par- 

 ticular case that the effect is to diminish the class by 1, and to 

 give rise to an inflexion, the stationary tangent being in fact 

 the common tangent of the curve and the absolute : I assume 

 that this is the case generally. Suppose that there are 6 con- 

 tacts, then there will be a diminution = 6 in the class, or this 

 will be =m 2 — 28— 3«— ; and there will be 6 inflexions ; there 

 may however be special circumstances giving rise to fresh in- 

 flexions, and I will therefore assume that the number of inflexions 

 is =tl, 



2 A2 



