673] 
259 
673. 
NOTE ON THE THEORY OF CORRESPONDENCE. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), 
pp. 32, 33.] 
If the point P on a given curve U of the order m, and the point Q on a 
given curve V of the order m', have a (1, 1) correspondence, this implying that 
the two curves have the same deficiency; then if PQ intersects the consecutive line 
P'Q' in a point P, the locus of R is a curve W of the class m + m', and the point 
R on this curve has, in general (but not universally), a (1, 1) correspondence with 
the point P on U or with the point Q on V. For, considering the correspondence 
of the points P and R, to a given position of P there corresponds, it is clear, a 
single position of R; on the other hand, starting from R, the tangent at this point 
to the curve W meets the curve U in m points and the curve V in m' points, but 
it is in general only one of the m points and only one of the m' points which are 
corresponding points on the curves TJ and V; that is, it is only one of the m points 
which is a point P; and the correspondence of (P, R) is thus a (1, 1) correspondence. 
But the curves TJ, V may be such that the correspondence of (P, R) is not a 
(1, 1) but a (Jc, 1) correspondence; viz., that to a given position of P there 
corresponds a single position of R, but to a given position of R, k positions of P. 
To show that this is so, imagine through P a line II having therewith a (k, 1) 
correspondence; P being, as above, a point on the curve TJ, the line in question 
envelopes a curve W\ and the correspondence is such that, for any given position 
of P on the curve TJ, we have through it a single position of the line: but, for a 
given tangent of the curve W, we have upon it k positions of the point P, viz. k 
of the m intersections of the line with the curve TJ are points corresponding to the 
line; this, of course, implies that the curve TJ is not any curve whatever of the 
order m, but a curve of a peculiar nature. 
33—2