146 
PROFESSOR CAYLET’S SECOND MEMOIR ON THE 
above referred to. The fundamental theorem in regard to unicursal curves, viz. that in 
a curve of the order m with \{m— l){m— 2) double points (nodes or cusps) the coordi- 
nates (#, y, z ) are proportional to rational and integral functions of a variable parameter 
Q , — as a case of a much more general theorem of Riemanns — dates from the year 1857, 
but was first explicitly stated by Clebsch in the paper “ Ueber diejenigen ebenen Curven 
deren Coordinaten rationale Functionen eines Parameters sind,” Crelle, t. 64 (1864), pp. 
43-63. See also my paper “ On the Transformation of Plane Curves,” London Mathe- 
matical Society, No. III., Oct. 1865. 
The paragraphs of the present Memoir are numbered consecutively with those of the 
First Memoir. 
On the Correspondence of two points on a Curve . — Article Nos. 94 to 104. 
94. In a unicursal curve the coordinates (%, y, z) of any point thereof are proportional 
to rational and integral functions of a variable parameter 0. Hence if two points of 
the curve correspond in such wise that to a given position of the first point there corre- 
spond a' 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 For let the two points be determined by their parameters Q , S' respectively, 
then to a given value of S there correspond a! values of S', and to a given value of S 
there correspond a values of S ; hence the relation between (S, S') is of the form 
(S, 1)“(S', l) a '=0; and writing therein S'=S, then for the points which correspond each 
to itself, we have an equation (S, l) a+a '— .0, of the order a-\-a! ; that is, the number of 
these points is =a-f-a'. 
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!. But a unicursal curve is nothing else than a curve with a deficiency 
D = 0, and we thence infer — 
Theorem. If two points of a curve with deficiency D have an (w, a!) correspondence, 
the number of united points is =«-}-a'-(-2/M) ; in which theorem 2 k is a coefficient to 
be determined. 
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 0 (the 
equation of the curve 0 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 
0 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 -J- 
whence in particular if the curve 0 does not pass through the point P, then the number 
of united points is = a-f- a', as in the case of a unicursal curve. (I have in the paper of 
