126: 
For, on account of the uniform continuity of the correspondence 
between J and B, to a sequence of points of JZ possessing only 
one limiting point, a sequence of points of & likewise possessing 
only one limiting point, must correspond, and reciprocally. On this 
ground the given correspondence already admits of an extension to 
a one-one transformation of the cube with its boundary in itself of 
which we have still to prove the continuity in the property that a 
sequence {gj of limiting points of M converging to a single limiting 
point Jno, the sequence {g,,} of the corresponding limiting points of 
R converges likewise to a single limiting point. For this purpose we 
adjoin to each point gn, a point m, of M possessing a distance 
<e from g,,, the distance between g,, and the point 7, corresponding 
to m, likewise being < «,, and for v indefinitely increasing we make 
e, to converge to zero. Thus {m,} converging exclusively to gna, {7} 
likewise possesses a single limiting point g,.,, and also {g,,} must 
converge exclusively to g,.. 
On account of the invariance of the number of dimensions *) we 
can enunciate as a corollary of theorem 10: 
TureorEM 11. For m<n the geometric types 4” and n" are different. 
As, however, for normally connected sets in general the notion 
of uniform continuity is senseless, the udeterminateness of the number 
of dimensions of everywhere dense, countable, multiply ordered sets, 
as expressed in theorem 9, must be considered as irreparable. 
Mathematics. — “An involution of associated points.” By Prof. JAN 
DE VRIES. 
(Communicated in the meeting of February 22, 1913). 
§ 1. We consider three pencils of quadric surfaces (a?), (0°), (c®”), 
the base curves of which may be indicated by a’, p*, y*. By the 
intersection of any surface a? with any surface 6? and any surface 
c* an mvolution of associated points, L*, consisting of oo* groups, is 
generated. Any point outside a‘, p*, y* determines one group. 
Through any point A of «* passes one surface 4° and one surface 
c’; these quadrics have a twisted quartic (A)* in common, intersected 
by the surfaces of pencil (a?) in oo groups of seven points A’ 
completed by A to groups of the /*. The points of the three base 
curves are singular. 
1) Comp. Math. Annalen 70, p. 161. 
