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