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SECOND MEMOIR ON THE 
[407 
gives a point which reckons as j united points. But if 6=X gives the J-fold root 
6' = X, this shows that the line 6 = X has with the curve j intersections at the point 
6 = 6'= X; not that the line 6=6' has with the curve j intersections at the point in 
question. 
98. Reverting to the notion of a united point as a point P which is such that one 
or more of the corresponding points P' come to coincide with P; in the case where 
P is at a node of the given curvé, it is necessary to explain that the point P must 
be considered as belonging to one or the other of the two branches through the node, 
and that the point P is not to be considered as a united point unless we have on 
the same branch of the curve one or more of the corresponding points P' coming to 
coincide with the point P. If, to fix the ideas, & = 1, that is, if the curve © simply 
pass through the point P, then if P be at a node the curve © passes through the 
node and has therefore at this point two intersections with the given curve; but the 
second intersection belongs to the other branch, and the node is not a united point; 
in order to make it so, it is necessary that the curve © should at the node touch 
the branch to which the point P is considered to belong. The thing appears very 
clearly in the case of a unicursal curve; we have here two values 6 = X, 6 = X' 
answering to the node according as it is considered as belonging to one or the other 
branch of the curve; and in the equation of correspondence (6, l) a (6\ 1)° =0, writing 
6 = X, we have an equation (X, 1 ) a (6', l) a '=0 satisfied by 6'= X' but not by 6'=X, and 
the equation (6, 1 ) a (6, 1)“' = 0 is thus not satisfied by the value 6=X. The conclusion 
is that a node qua node is not a united point. 
99. But it is otherwise as regards a cusp. When the point P is at a cusp, the 
curve © (which has in general with the given curve k intersections at P) has here 
more than k intersections, and (as in this case there is no distinction of branch) the 
cusp reckons as a united point. In the case of a unicursal curve, there is at the cusp 
a single value 6=X of the parameter, and the equation (6, l)“(d, l) a =0 is satisfied 
by the value 6 = X. But for the very reason that the cusp qua cusp reckons as a 
united point, the cusp is a united point only in an improper or special sense, and it 
is to be rejected from the number of true united points. We may include the cusps, 
along with any other special solutions which may present themselves, under a head “ Supple 
ment,” and instead of writing as above a — a — oí = 2kD, write a — a — a' + Supp. = 2kD. 
Before going further I apply the theorem to some examples in which the curve 
© is a system of lines. 
100. Investigation of the class of a curve of the order m with 8 nodes and k 
cusps. Take as corresponding points on the given curve two points such that the line 
joining them passes through a fixed point 0; the united points will be the points 
of contact of the tangents through 0 ; that is, the number of the united points will 
be equal to the class of the curve. The curve © is here the line OP which has with 
the given curve a single intersection at P; that is, we have &=1. The points P' 
corresponding to a given position of P are the remaining m — 1 intersections of OP 
with the curve, that is, we have a! = m — 1; and in like manner a. =m—l. Each of the