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munication I propose first to prove this result by the method of 
correspondence alone, and then to extend the proof to higher 
orders of correspondence. 
In a (2, 2) correspondence there are 4 (= 2+ 2) united points, 
that is to say, four points each of which coincides with one of 
its correspondents. In fact, if two numbers # and y are con- 
nected by an equation of the second degree in each of them, 
then when we make w and y coincide, there results an equation 
of the fourth degree (Chasles, Comptes Rendus, 1864). I call 
these united points the points a. Each point a has one of its 
correspondents coinciding with it; it has also another corre- 
spondent b. Each point b again has another correspondent c, and 
soon. There are also four points a, each of which is such that 
its two correspondents coincide in a point ®. For let qg be a 
correspondent of p and r a correspondent of q; then the relation 
between p and r is a (2, 2) correspondence (since to each position 
of p there are two positions of r and vice versa), and therefore 
has four united points, viz. the points 8. Each of these points 
8 has another correspondent y, and so on. We have thus two 
series of points, abcd... aBy5... each letter indicating a set of 
four generally distinct points. 
Let us now endeavour to obtain a closed cycle of an odd 
order: for distinctness’ sake we will try to draw a pentagon 
msceribed in one conic, U, and circumscribed to another, V. 
Start with a point « on the outer ; pass to one of its correspond- 
ents, y; y has another correspondent, z; from z we go to u, 
from u to v, from v tow. If now w were the same point as a, 
we should have succeeded in our object. But the relation be- 
tween w and «# is a (2, 2) correspondence, for we might have 
started from « in either of two directions. The united points 
of this correspondence should therefore apparently give solutions 
of our problem. 
But these united points are no other than the four points c, 
For starting with one of these, we get the cycle cbaabe, which 
