395] 
INVESTIGATIONS IN CONNEXION WITH CASEY’S EQUATION. 
67 
each oi the corresponding points on J; and each cusp of 2 corresponds to two coincident 
points of J, viz. the point (/, g, h) being at a cusp of 2, the curve /U + gV+hW = 0 
is a cuspidal curve having a cusp at the corresponding point of J. The number of 
the binodal cuives fU + g\ + ATF = 0 is thus equal to the number of the nodes of 2, 
and the number oi the cuspidal curves fU-\-gV+hW = Q is equal to the number of 
the cusps of 2. Lhe curve 2 is easily shown to be a curve of the order 3 (n—l) 2 
and class 3n (n — 1); and qua curve which corresponds point to point with J, it is a 
curve having the same deficiency as J, that is a deficiency =£(3n-4)(3n-5); we 
have thence the Pliickerian numbers of the curve 2, viz.: 
Order is 
Class 
Cusps 
Nodes 
Inflexions 
= o(n — l) 2 , 
= 3n (n - 1), 
= 12 (n — 1) ( n — 2), 
= f (w - 1) ( n - 2) (S?i 2 - 3n - 11), 
= 3 (n — 1) (4n — 5), 
Double tangents = § (n -1) ( n — 2) (3n 2 + 3n - 8). 
Remai'ks. The consideration of the foregoing curve 2 is, I believe, first due to 
Prof. Cremona, it is a curve related to the three distinct curves U = 0, V = 0, W = 0, 
in the same way precisely as Steiner’s curve P 0 is related to the three curves 
(4/7=0, d y U =0, (4/7=0. (Steiner, “ Allgemeine Eigenschaften der algebraischen Curven,” 
Crelle, t. xlvii. (1854), pp. 1—6 ; see also Clebsch, “ Ueber einige von Steiner behandelte 
Curven,” Crelle, t. lxiv. (1865), pp. 288—293), and the Pliickerian numbers of P 0 
(writing therein n -f 1 for n) are identical with those of 2. The foregoing expressions 
|(?i — 1) ( n — 2) (3/i 2 — 3n — 11) and 12 (n — l)(w — 2) for the numbers of the binodal and 
cuspidal curves fU + gV + hW = 0, are given in my memoir “On the Theory of Invo 
lution,” Cambridge Philosophical Transactions, t. xi. (1866), pp. 21—38, see p. 32, [348] \ 
but the employment of the curve 2 very much simplifies the investigation. 
Passing now to the proposed question, we have as before the curves JJ—0, V=0, W=0, 
of the same order n; and we may consider the point (/, g, h), and corresponding thereto 
the curve \J (fU) + V (g V) + V (hW) = 0, say for shortness the curve 12, which is a curve 
of the order 2n, having n 2 contacts with each of the given curves U, V, W. As long 
as the point (/, g, h) is arbitrary, the curve 12 has not any node; and in order that 
this curve may have a node, it is necessary that the point (f, g, h) shall lie on a 
certain curve A ; this being so, the node will lie on the foregoing curve J, the Jacobian 
of the given curves U, V, W; and the curves J and A will correspond to each other, 
point to point, viz. taking for (f, g, h) any point whatever on the curve A, the curve 
12 will have a node at some one point of «7; and conversely, in order that the curve D 
may be a curve having a node at a given point of J, it is necessary that the point 
(f, g, h) shall be at some one point of the curve A. lhe curve A has howevei nodes 
and cusps; each node of A corresponds to two points of J, viz. foi (f, g, h) at a node 
of A, the curve 12 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. foi (f g, h) 
at a cusp of A the curve 12 is a cuspidal curve having a cusp at the corresponding 
9—2