10 
ON THE CORRESPONDENCE OF TWO POINTS ON A CURVE. 
[385 
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 ¿ intersections at the point P, 
then in the system of (P, P'), the number of united points is 
a = a + a' + 2 kD, 
whence in particular, if the curve © does not pass through the point P, then the 
number of united points is = a + a, as in a unicursal curve. 
4*. The foregoing theorem is easily proved in the particular case where the k 
intersections at the point P take place in consequence of the curve 6 having a ¿-tuple 
point at P. Taking U = (x, y, z) m = 0 as the equation of the given curve (which for 
greater simplicity is assumed to be a curve without singularities), then if we suppose 
{x, y, z) to be the coordinates of the point P, and {x\ y, z') to be the coordinates 
of the point P', write U = (x, y, z) m , U' = (x', y', z') m , U' being what U becomes on 
writing therein (x' } y, z') in place of (x, y, z); and 
© = (x, y, z) a (x', y', z'Y (yz' - y'z, zx - z'x, 
xÿ - x'yf, 
viz., © is a function of the order k in yz — yz', zx — zx, xy — xy, the coefficients of 
the several powers and products of these quantities being functions of the order a in 
(x, y, z) and of the order a' in (of, y', z'), which functions are such that they do not 
all of them vanish, identically, or in virtue of the equation TJ = 0, on writing therein 
(x', y', z) = (x, y, z). Taking for a moment (x, y, z) as current coordinates, suppose 
that the equation of the given curve is U= 0; then if (x, y, z) are the coordinates 
of the point P, we have U — 0, and similarly if (x', y, z') are the coordinates of the 
point P' we have U' = 0. The equation © = 0, considering therein (x, y, z) as the 
coordinates of the given point P (and so as parameters satisfying the equation U — 0) 
and {pc’, y', z') as current coordinates, will be a curve having a ¿-tuple point at P, 
we have thus the case above supposed; and P being given, the corresponding points 
P' are given as the intersections of the curves U' = 0, © = 0, which are respectively 
of the orders in and a' + k; the total number of intersections is thus = m (a' + k), 
but inasmuch as the curve © = 0 has a ¿-tuple point at P, k of these intersections 
coincide with the point P, and the number of the remaining intersections, that is 
the number of positions of the point P', is = ma! -f (m — 1) ¿. Similarly when P' is 
given, the number of positions of the point P is =ma + (m—l)k: and we have 
therefore 
a + cl = m {a + a') + 2 (m — 1) k. 
To find the united points, it is to be observed, that upon writing {x, y', z') = (x, y, z), 
the function © becomes identically = 0 ; but if we suppose, in the first instance, that 
P, P, are consecutive points on the curve U = 0, then we have 
yz — y'z : zx' — z'x : xy — x'y = U; S y U ; B Z U; 
and the equation © = 0 assumes the form 
© = 0> y, zf (x, y, z) a '(8 x U, ByU, 8 z ll) k = 0,