407]
CURVES WHICH SATISFY GIVEN CONDITIONS.
265
95. Suppose that the corresponding points are P, P' and imagine that when P
is given the corresponding points P' are the intersections of the given curve by a curve
(*) (the equation of the curve © will of course contain the coordinates of P as
parameters, for otherwise the position of P' would not depend upon that of P). I
find that if the curve © has with the given curve k intersections at the point P,
then in the system of points (P, P') the number of united points is
a = a. + a! + 2 kP,
whence in particular if the curve © does not pass through the point P, then the
number of united points is = a 4- a', as in the case of a unicursal curve. (I have in
the paper of April 1866 above referred to, proved this theorem in the particular case
where the k intersections at the point P take place in consequence of the curve ©
having a /¿-tuple point at P, but have not gone into the more difficult investigation
for the case where the k intersections arise wholly or in part from a contact of the
curve ©, or any branch or branches thereof, with the given curve at P.)
96. It is to be observed that the general notion of a united point is as follows :
taking the point P at random on the given curve, the curve © has at this point k
intersections with the given curve; the remaining intersections are the corresponding
points P'; if for a given position of P one or more of the points P' come to
coincide with P, that is, if for the given position of P the curve © has at this
point more than k intersections with the given curve, then the point in question is
a united point.
It might at first sight appear that if for a given position of P a number 2, 3,..
or j of the points P' should come to coincide with P, then that the point in question
should reckon, for 2, 3,... or j (as the case may be) united points: but this is not
so. This is perhaps most easily seen in the case of a unicursal curve; taking the
equation of correspondence to be (9, 1)“ {O', l) a ' = 0, then we have a+a' united points
corresponding to the values of 9 which satisfy the equation (9, l) a (9, 1)“’ = 0 ; if this
equation has a ^’-triple root 9 = \, the point P which answers to this value A of the para
meter is reckoned as j united points. But starting from the equation (9, 1)“ (91)“' = 0,
if on writing in this equation 9 = A, the resulting equation (A, 1)“ (9', 1)“ = 0 has a
root 9' = A, it follows that the equation (9, 1 ) a (9, l) a ' = 0 has a root 9 = A, and that
the point which belongs to the value 9 — A is a united point; if on writing in the
equation 9 = \, the resulting equation (A, 1)° (9', l) a ' = 0 has a y-tuple root 9' = A, it
does not follow that the equation (9, 1)° (9, 1)“'=0 has a y-tuple root 9 = A, nor con
sequently that the point answering to 9 = A in anywise reckons as j united points.
97. This may be further illustrated by regarding the parameters 9, 9' as the
coordinates of a point in a plane; the equation (9, 1 ) a (9 > , 1)“ = 0 is that of a curve
of the order a + a', having an a-tuple point at infinity on the axis 9 = 0, and an
a'-tuple point at infinity on the axis 9' = 0 ; the united points are given as the inter
sections of the curve with the line 9=9'; a j-fold intersection, whether arising from
a multiple point of the curve or from a contact of the line 9 = 9' with the curve,
c. vi. 34