264 
SECOND MEMOIR ON THE 
[407 
number of the proper solutions, I use these results to determine a 'posteriori in each 
case the expression of the Supplement; the expression so obtained can in some cases 
be accounted for readily enough, and the knowledge of the whole series of them will 
be a convenient basis for ulterior investigations. 
The Principle of Correspondence for points in a line was established by Chasles in 
the paper in the Comptes Rendus, June—July 1864, referred to in my First Memoir; it 
is extended to unicursal curves in a paper of the same series, March 1866, “ Sur les 
courbes planes ou a double courbure dont les points peuvent se determiner individuelle- 
ment—Application du Principe de Correspondance dans la theorie de ces courbes,” but 
not to the case of a curve of given deficiency D considered in my paper of April 1866 
above referred to. The fundamental theorem in regard to unicursal curves, viz. that in 
a curve of the order m with ^ (m — 1) (m — 2) double points (nodes or cusps) the 
coordinates (x, y, z) are proportional to rational and integral functions of a variable 
parameter 6,—as a case of a much more general theorem of Riemann’s—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. LXiv. (1864), pp. 43—63. See also my paper “ On the Transformation of Plane 
Curves,” London Mathematical Society, No. III., Oct. 1865, [384]. 
The paragraphs of the present Memoir are numbered consecutively with those of 
the First Memoir. 
Article Nos. 94 to 104.—On the Correspondence of two points on a Curve. 
94. In a unicursal curve the coordinates (x, y, z) of any point thereof are pro 
portional to rational and integral functions of a variable parameter 6. Hence if two 
points of the curve correspond in such wise that to a given position of the first point 
there correspond ol 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 6, 6' 
respectively, then to a given value of 6 there correspond a! values of 0\ and to a 
given value of 6' 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. 
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 :