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SECOND MEMOIR ON THE
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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