Mathematics. — “/nvolutorial Correspondences (2,2) of the Hurst 
Class’. By Prof. Jan pe Vries. 
(Communicated at the meeting of April 30, 1921). 
§ 1. An involutorial correspondence (2,2) of the jirst class is 
characterized by the property that an arbitrary straight line contains 
one pair of associated points P,P*. If we associate to each other 
the straight lines joining a point / to the two homologous points 
P, and P,, also the field of rays is arranged in an involutorial (2,2). 
At the same time there arises a null system, if we associate to P 
the straight lines PP, and PP,; each straight line has in this case 
two null points, each point has two null rays. 
If the point P describes the straight line 7, its null rays envelop 
a curve ( of the fourth class that has 7 as a double tangent. The 
six points V in which (7), is cut by 7, are evidently branch points 
of the (2,2). The branch curve (FV) of the (2,2) is therefore a curve 
of the order sz. 
We shall now suppose that the locus of the coincidences P= P* 
is a curve of the order n. If P describes the line #, the points P,, 
P, associated to P, describe a curve e, which has the x coincidences 
on 7 and the pair of associated points on 7, P,P*, in common with r. 
Through this correspondence r is therefore transformed into a 
curve oft? of the order (n+2). 
Let us now consider the curves 9,”+? and o,”+? corresponding to 
the straight lines 7, and 7,. Besides the two points associated to 
S= rr, they have the points P in common for which P, lies on 
r, and P, on r,; the other common points are singular, i.e. each 
of them is associated to oc! pairs P,, 1,. 
The curves (r,), and (r,), corresponding to the straight lines r, 
and r,, have in the first place the two null rays of the point S in 
common. The line r, cuts 0,"+? in (n-+2) points P,, which are 
associated to as many points P, on 7, and accordingly define (n+ 2) 
common tangents. The other (12—n) common tangents are evidently 
singular straight lines; each of them bears oo! pairs of points P, P*. 
Let us also consider the locus of the pairs of points P, P* which 
are collinear with a point O. Let O, and O, be the points conjugated 
to O through (2,2); the curve in consideration w is touched at O 
