385] 
9 
385. 
ON THE CORRESPONDENCE OF TWO POINTS ON A CURVE. 
[From the Proceedings of the London Mathematical Society, vol. i. (1865—1866), No. vu. 
pp. 1—7. Read April 16, 1866.] 
1. In a unicursal curve the coordinates (x, y, z) of any point of the curve are 
proportional to rational and integral functions of a variable parameter 6. Hence, if 
two points of the curve correspond in suchwise that to a given position of the first 
point there correspond oi positions of the second point, and to a given position of 
the second point a positions of the first point, the number of points which correspond 
each to itself is =a + a'. For let the two points be determined by their parameters 
0, 6' respectively—then to a given value of 6 there correspond a' values of O', and 
to a given value of 0' there correspond a values of 0 ; hence the relation between 
(0, 0') is of the form (0, l) a (0', l) a ' = 0; and writing therein 0' = 0, then for the points 
which correspond each to itself, we have an equation (0, l) a+a '= 0 of the order a + a'; 
that is, the number of these points is = a + a'. 
2. Hence for a unicursal curve we have a theorem similar to that of M. Chasles’ 
for a line, viz., the theorem may be thus stated : 
If two points of a unicursal curve have an (a, a') correspondence, the number of 
united points is = a + a'. 
3. But a unicursal curve is nothing else than a curve with a deficiency I) = 0, 
and we thence infer 
Theorem. If two points of a curve with deficiency D have an (a, a') corre 
spondence, the number of united points is = a + a + 2kD ; in which theorem 2k is a 
coefficient to be determined. 
4. 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 
C. VI. 2