576 
ON POLYZOMAL CURVES. 
[414 
We have the curves £7=0, P"= 0, W = 0, each of the same order r; and con 
sidering a point the coordinates whereof are (l, m, n), we regard as corresponding to 
this point the curve VlU + VmD-f *JnW = 0, say for shortness, the curve if, being as 
above a curve of the order 2r, having r 2 contacts with each of the given curves 
£7 = 0, V = 0, W — 0. As long as the point (l, m, n) is arbitrary, the curve O has not 
any node, and in order that this curve may have a node, it is necessary that the 
point (l, 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 (l, m, n) any 
point whatever on the curve A, the curve il will have a node at some one point 
of J; and conversely, in order that the curve il may be a curve having a node at 
a given point of J, the point (l, 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 (l, m, n) at a node of A, the curve il 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 (l, m, n) at a cusp of A, the curve O has a node at 
the corresponding point of J. The number of the binodal curves il is thus equal to 
the number of the nodes of A, and the number of the cuspidal curves il 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 £ (viz., £ is the locus of 
the nodes of those curves lU+viV+nW=0 which have a node), and the result arrived 
at is that for the curve A 
Class = 6 (r — l) 2 , 
Nodes = f(i—1) (27r 2 — 63r 2 + 22r + 16), 
Cusps = 3 (r — 1) (7r — 8), 
Double tangents = f (r — 1) (12?^ — 36r 2 + 19r + 16), 
Inflexions 
= 12 (r — 1) (r — 2); 
so that, finally, the number of the cuspidal curves Vi£7+ *JmV+ \!nW — 0, is found to be 
= 3(r — 1) (7r —8), and the number of the binodal curves of the same form is found 
to be = |(r — 1) (27r 3 —63r 2 +22r + 16). When the given curves are conics, or for r = 2, 
these numbers are =18 and 36 respectively; but the formulae are not applicable to 
the case where the conics have a point or points of intersection in common; nor, 
consequently, to the case of the three circles.