Prof. Cayley on the Theory of the Evolute. 3 19 



and substituting in the equation 



we find 



OT'=2(n'-l) + (m-l)(m-2)-28-2*-i'; 



and the equation t'— /e' = 3(7i'— m') gives also 



«*=!— 3(pf— m^ + i't 



whence, attending to the value of n', we find the following sys- 

 tem of equations for the singularities of the evolute, viz. 



»'=!»*— 2 3-3 k- 6, 



m'=3m(m-l)-6 8-8 /c-20-i', 



i! = l\ 



^=3m(2m-3)-12S-15A;-36>-2^ 



and the values of t 7 and 8' may then also be found from the 

 equations 



m f =^(y-l)-2T , -3^ 



7^ / =mV-l)-28'-3K , . 



I have given the system in the foregoing form, as better exhi- 

 biting the effect of the inflexions; but as each of the 6 contacts 

 with the absolute gives an inflexion, we may write I' — d + i 11 , 

 where, in the absence of special circumstances giving rise to any 

 more inflexions, t! 1 = 0. The system thus becomes 



n'=m 2 -2 8-3 k- 6, 



m' = 3m(m-l)-6 8-8 fc-Sd-i", 



•'= e+t,", 



k' = 37»(2ro-3)-128-15*— 50-2*", 

 so that each contact with the absolute diminishes the class by 1, 

 the order by 3, and the number of cusps by 5. 



I remark that when the absolute becomes a pair of points, a 

 contact of the given curve m means one of two things : either 

 the curve touches the line through the two points, or else it 

 passes through one of the two points : the effect of a contact of 

 either kind is as above stated. Suppose that the two points are 

 the circular points at infinity, and let m = 2, the evolute in ques- 

 tion is then the evolute of a conic, in the ordinary sense of the 

 word evolute. We have, in general, class =4, order =6; but 

 if the conic touches the line infinity (that is, in the case of the 

 parabola), the reductions are 1 and 3, and we have class =3, 

 order =3, which is right. If the conic passes through one of 

 the circular points of infinity, then in like manner the reduc- 

 tions are 1 and 3 ; and therefore if the conic passes through each 

 of the circular points at infinity (that is, in the case of a circle), 



