CURVES WHICH SATISFY GIVEN CONDITIONS. 
147 
April 1866 above referred to, proved this theorem in the particular case where the Jc 
intersections at the point P take place in consequence of the curve 0 having a £-tuple 
point at P, but have not gone into the more difficult investigation for the case where 
the Jc 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 0 has at this point Jc 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 0 has at this point more than Jc 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, . . ox 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 (0, 1)“(0', l) a '=0, then we have a-\-a! united points corresponding to the values of 
0 which satisfy the equation (0, l) a (0, 1)“'=0; if this equation has a /-tuple root 0= A, 
the point P which answers to this value X of the parameter is reckoned as / united points. 
But starting from the equation (0, l) <x (0 , , l) a = 0, if on writing in this equation 0=X, the 
resulting equations (X, 1)“(0 / , l) a '=0 has a root 0'=X, it follows that the equation 
(0, 1)“(0, l) a '=0 has a root 0= A, and that the point which belongs to the value 0=X is 
a united point ; if on writing in the equation 0=X, the resulting equation (a, l) a (0', l) a '=0 
has a /-tuple root 0'=X, it does not follow that the equation (0, 1)“(0, l) a '=0 has a /-tuple 
root 0=X, nor consequently that the point answering to 0=X in anywise reckons as 
/ united points. 
97. This may be further illustrated by regarding the parameters 0, 0' as the coordinates 
of a point in a plane ; the equation (0, 1)“(0', l) a '=0 is that of a curve of the order a -fa', 
having an a-tuple point at infinity on the axis 0 = 0, and an a'-tuple point at infinity on 
the axis 0' = O; the united points are given as the intersections of the curve with the 
line 0=0'; a /-fold intersection, whether arising from a multiple point of the curve or 
from a contact of the line 0=0' with the curve, gives a point which reckons as / united 
points. But if 0=X gives the /-fold root 0'=X, this shows that the line 0=X has with 
the curve / intersections at the point 0 = 0'= A ; not that the line 0=0/ has with the curve 
/ 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 curve, 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, Jc= 1, that is, if the curve 0 simply pass through the point P, then if 
mdccclxviii. ‘ y 
