350 Prof. Cayley on the Theory of the Evolute. 



the reductions are 2 and 6, and we have class = 2, order =0, 

 which is also right; for the evolute is in this case the centre, 

 regarded as a pair of coincident points. That this is so, or that 

 the class is to be taken to be (not = 1 but) = 2, appears by the 

 consideration that the number of normals to the circle from 

 a given point is in fact = 2, the two normals being, however, 

 coincident in position. 



To complete the theory in the general case where the abso- 

 lute is a proper conic, I remark that, besides the inflexions which 

 arise from contacts of the given curve with the absolute, there 

 will be an inflexion, first, for each stationary tangent of the 

 given curve which is also a tangent of the absolute; secondly, 

 for each cusp of the given curve situate on the absolute. Hence, 

 if the number of such stationary tangents be =X, and the num- 

 ber of such cusps be =/*, we may write */' = \-f-/^ and there- 

 fore also i l = + \ + fi. 



I remark also that we have 



— 28— 3k=— m(m—l) + n, 



-6S-8*=-3m(m-2)-H, 

 and therefore also 



-128-15*= -6m 2 + 15w-3ra + 3i. 

 The general formulas thus become 



n' sa m + n — , 



m'=3m + 1—20— i\ 



<! = i\ 



k! =6m-3rc+3t-30-2t'. 

 If instead of the given curve we consider its reciprocal in 

 regard to the absolute, then 



m, n, 8, k, t, i ; 0, X, fi; t' = + \ + fM 

 are changed into 



n, m, t, Cj 8, k; 0, fjbj\; t!=6 + fi + \ 

 respectively. And for the evolute of the reciprocal curve we have 

 n 1 = n+ m — 6 J, 

 m' = Sn + K-20-i', 



i! m i\ 



h! = 6n-3m + 3fc-30-i', 



which, attending to the relation i — /c=3(n — m), are in fact the 

 same as the former values ; that is, the evolute of the given 

 curve, and the evolute of the reciprocal curve are curves of the 

 same class and order, and which have the same singularities. 

 Cambridge, February 22, 1865. 



