385] 
ON THE CORRESPONDENCE OF TWO POINTS ON A CURVE. 
11 
which, (%, y, z) being current coordinates, is the equation of a curve of the order 
a+a + (to —1)&; the united points are the intersection of this curve with the given 
curve U — 0, and the number of the united points is thus ^ 
a = in (a + a!) + m {m — 1) k. 
Hence attending to the above-mentioned value of a + a', we have 
a = a + a! + (to — 1) (to — 2) k. 
But in the case of a curve U = 0, without singularities, we have 2D = (to — 1) (to — 2), 
and we have thus the required formula 
a = a + a' + 2 kD. 
The investigation in the case where the k intersections at P arise wholly or in 
part from a contact of the curve ©, or any branch or branches thereof, with the 
given curve U, is more difficult, and I abstain from entering upon it. 
I apply the theorem to some examples: 
5. Investigation of the class of a curve of the order to with 8 dps. Take as 
corresponding points on the 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 is 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 to — 1 intersections of OP with the curve; 
that is, we have a' = to — 1; and in like manner a = to — 1. Hence the class is 
= 2 (to— 1) + 2P, viz., this is = (2m — 2) + (to 2 — 3to + 2 — 28), which is = to 2 — to — 28, as 
it should be. 
6. It is in the foregoing example assumed that the dps are none of them cusps; 
if the curve has 8 + k dps, k of which are cusps (or what is the same thing, 8 nodes 
and k cusps); then the number of united points is equal 2 (to — 1) + 2D, = to 2 — to — 28 — 2k ; 
but in this case each of the cusps is reckoned as a united point, and we have, therefore, 
class -f k = to 2 - m - 28 - 2k, that is, class = to 2 -to - 28 - 3«. This will serve as an 
instance of the special considerations which are required in the case of a curve with 
cusps, but in what follows, I shall assume that the dps are none of them cusps and 
thus attend to the case of a curve of the order to, with 8 dps, and therefore of the 
class n = m 2 — in — 28, and of the deficiency D = \ (m - 1) (to — 2) — 8, = \ (n - 2to + 2). 
7. Investigation of the number of inflexions. Taking the point P' to be a 
tangential of P (that is, an intersection of the curve by the tangent at P), the 
united points are the inflexions, and the number of the united points is equal to the 
number of inflexions. The curve © is here the tangent at P, having with the given 
curve two intersections at P; that is k = 2. P' is any one of the to — 2 tangentials 
of P, hence a = m - 2; and P is the point of contact of any one of the n - 2 
2—2