PROFESSOR CAYLEY ON POLYZOMAL CURVES. 109 



however seen how the ratios / : m : n may be determined so that the curve shall 

 acquire a node, two nodes, or a cusp ; viz., regarding a, b, c as current areal co- 

 ordinates, we have here a conic + t + - = 0, the locus of the centres of the 



a b c ' 



variable circle, and the solution depends on establishing a relation between this 

 conic and the orthotomic circle or Jacobian of the three given circles. I have in 

 my paper ' : Investigations in connection with Casey's Equation," Quart. Math. 

 Jour. vol. viii. (1867), pp. 334-342, given, after Professor Cremona, a solution of 

 the general question to find the number of the curves Juj + JmV + s/n W= 0, 

 which have a cusp, or which have two nodes, and I will here reproduce the 

 leading points of the investigation. I remark, that although one of the loci 

 involved in it is the same as that occurring in the case of the three circles (viz., 

 we have in each case the Jacobian of the given curves), the other two loci 

 2 and A, which present themselves, seem to have no relation to the conic of 

 centres which is made use of in the particular case. 



We have the curves U= 0, V— 0, W— 0, each of the same order r ; and 

 considering a point the co-ordinates whereof are (I, m, n), we regard as corres- 

 ponding to this point the curve / JfU+ JmV+ JnW= 0, say for shortness, the 

 curve Q, being as above a curve of the order 2r, having r 2 contacts with each of 

 the given curves U — 0, V= 0, W= 0. As long as the point (/, m, n) is arbitrary, 

 the curve Q has not any node, and in order that this curve may have a node, it is 

 necessary that the point (I, m, n) shall lie on a certain curve A ; this being so, the 

 node will, it is easy to see, lie on the curve J, the Jacobian of the three given 

 curves ; and the curves J and A will correspond to each other point to point, 

 viz., taking for (I, m, n) any point whatever on the curve A, the curve Q will have 

 a node at some one point of J; and conversely, in order that the curve Q may 

 be a curve having a node at a given point of J, the point (/, m, n) must be at 

 some one point of the curve A. The curve A has, however, nodes and cusps; each 

 node of A corresponds to two points of J, viz., for (/, m, n) at a node of A, the 

 curve Q is a binodal curve having a node at each of the corresponding points of J; 

 each cusp of A corresponds to two coincident points of J, viz, for (I, m, n) at a cusp 

 of A, the curve Q has a node at the corresponding point of J. The number of the 

 binodal curves Q is thus equal to the number of the nodes of A, and the number 

 of the cuspidal curves Q is equal to the number of the cusps of A ; and the 

 question is to find the Pliickerian numbers of the curve A. This Professor 

 Cremona accomplished in a very ingenious manner, by bringing the curve A into 

 connexion with another curve 2 (viz., 2 is the locus of>the nodes of those curves 



VOL. XXV. PART I. 2 E 



