348 Prof. Cayley on the Theory of the Evolute* 



Suppose in general that for any curve we have 

 m } the order, 

 n, „ class, 

 8, „ number of nodes, 

 k, „ „ cusps, 



r, „ „ double tangents, 



ij „ „ inflexions. 



Then PKicker's equations give 



i— K = 3(n— m)j T—8=i(n—m)(n-i-m — 9); 

 and we thence have ■ . 



6-* + T-S=i(n-.l)(n-2)-i(«i-l)(m-2),- 

 .or, what is the same thing, 



i(m-l)(m~2)--S-/c=i(7i-l)(n-2)-T-t. 



Now M. Clebsch in his recent paper " Ueber die Singulari- 

 taten algebraischer Curven," Crelle, vol. Ixiv. (1864) pp. 98-100, 

 has remarked (as a consequence of the investigations of Riemann 

 in the Integral Calculus) that whenever from a given curve 

 another curve is derived in such manner that to each point (or 

 tangent) of the given curve there corresponds a single tangent 

 (or point) of the derived curve, then the expression 



i(m-l)(m-2)— 8— *, = £(» — l)(n — 2)-t— i, 



has the same value in the two curves respectively, or that, writing 

 m\ n ! , hf, «', T r , t! for the corresponding quantities in the second 

 curve, then we have 



i(m-l) (m-2) -8-* = i{n-l) (n-2)— r-* 

 = J(m f -l)(m / -2)-3'-^=i(^-l)(^-2)-T'-^ 



and consequently that, knowing any two of the quantities m!, n\ 

 £', k\ t', d, the remainder of them can be determined by means 

 of this relation and of Plucker's equations. The theorem is 

 applicable to the evolute according to the foregoing generalized 

 definition* ; and starting from the values 



7i' = m 2 -2S-3fl;-0, 



we find in the first instance 



T ' = ±( n '-l)(n'--2)--i(m-l)(m-2) + $ + fc-i f ; 



* M. Clebsch in fact applies it to the evolute in the ordinary sense of the 

 term, but by inadvertently assuming t'=& instead of t=0 he is led to some 

 incorrect results. 



