260 
NOTE ON THE THEORY OF CORRESPONDENCE. 
[673 
Imagine now that we have on the line II a point Q, having with P a (1, 1) 
correspondence of a given nature: to fix the ideas, suppose P, Q are harmonics in 
regard to a given conic: since on each of the lines II there are k positions of P, 
there are also on the line k positions of Q, and the locus of these k points Q is a 
curve V, say of the order m'. 
The point P on the curve U and the point Q on the curve V have a (1, 1) 
correspondence. For, consider P as given: there is a single position of the line II 
intersecting V in m' points, but obviously only one of these is the point Q. And 
consider Q as given: then through Q we have say ¡x tangents of the curve W; each 
of these tangents intersects the curve U in m points, k of which are points P, but 
for a tangent taken at random no one of these is the correspondent of Q; it is, in 
general, only one of the ¡x tangents which has upon it k points P, one of them 
being the point corresponding to Q; that is, to a given position of Q there corresponds 
a single position of P; and the correspondence of the points (P, Q) is thus a (1, 1) 
correspondence. 
We have thus the point P on the curve U and the point Q on the curve V, 
which points have with each other a (1, 1) correspondence ; and the line II is the 
line PQ joining these points; this intersects the consecutive line in a point R; and 
the locus of R is the curve W. To a given position of P there corresponds a single 
line II, and therefore a single position of R; but to a given position of R there 
correspond k positions of P, viz. drawing at R the tangent to the curve W, this is 
a line II having upon it k points P, or the correspondence of (P, Q) is, as stated, 
a (k, 1) correspondence. 
The foregoing considerations were suggested to me by the theory of parallel 
curves. Take a curve parallel to a given curve, for example, the ellipse; this is a 
curve of the order 8, such that every normal thereto is a normal at two distinct 
points; and the curve has as its evolute the evolute of the ellipse, or, more 
accurately, the evolute of the ellipse taken twice; but, attending only to the evolute 
taken once, each tangent of the evolute is a normal of the parallel curve at two 
distinct points thereof, and the points of the parallel curve have with those of the 
evolute not a (1, 1) but a (2, 1) correspondence.