Colloquium publications.

THE BOSTON COLLOQUIUM.

analogues of +-curves would be, in space, 0-surfaces, and that
those that pass through a point P on any curve of the system
would contain its conjugate Q; and further, that all points Q
conjugate to P would lie on some determinate curve of the surface. This last supposition can be established by reductio ad
absurdum.
The surface F and its plane sections (C) are algebraic. Each
curve C containing a selected point P has, by hypothesis, one particular point Q conjugate to P. Therefore the Q's on the oo2
curves through P must suit one of the following three descriptions.
(a) They may be finite in number, Q1, Q2, - Q,. But then
every plane section of the surface through P would need to contain the line PQ1, or PQ,, ', or PQ,. This is absurd.
(b) The Q's may be simply infinite, ool in number, filling one
or more algebraic curves on the surface, or lastly
(c) The Q's may fill all the oC2 points of the surface F. We
shall reject this after showing that in this case the surface must
be rational; i. e., rationally and reversibly transformable into a
plane, whereas on the contrary, in a plane or any other rational
surface the Q's must be only a simple infinity, ool.
Suppose, therefore, that every point Q of the surface is conjugate to a given P upon some one or more curves of the system.
It cannot be so upon all plane sections through the secant PQ, for
then must every point of any plane section be conjugate to P on
that section, contrary to the hyperelliptic hypothesis. Through
every ray PQ there lie then a finite number r of planes in which
P and Q are conjugate. Any one of these determines all the others,
for P and the plane through P fix Q, and the rest follow. Now
such a grouping of the planes through P into sets of r planes, each
set being determined by any one of its planes, is called an involution. Castelnuovo * proves the remarkable theorem, that every
involution of the planes about a point in space of three dimensions
is rationl; i. e. its groups can be correlated univocally and
reversibly to the points of a plane, each group to one point.
* Math. Annalen, vol. 44 (1894), pp. 125-155.