Mathematics. — “An Involutory Transformation of the Rays of 
Space which is defined by two Involutory Homologies.” By 
Prof. JAN DE VRIES. 
(Communicated at the meeting of February 22, 1919). 
1. In a plane « I consider the involutory homology (central colli 
neation) which has A for centre and a for axis, in a plane 9 a 
similar involution with centre B and axis 4. If P,P’ is a pair of 
the first involution, Q, Q’ a pair of the second, | associate the rays 
1= PQ and tt =P’Q. In this way arises an involution in the 
rays of space, which will be ‘investigated in what follows. 
When PQ and P’Q’ intersect in a point M, the pair Q,Q’ is the 
central projection of P,P’ out of M as centre. By means of this 
projection the pairs of the involution [«] lying on p= PP’ are 
transformed into the pairs of an involution situated on ¢= QQ’; 
the latter has one pair in common with the involution which is 
defined on q by the homology [8]. Consequently through M passes 
one pair of rays tf’. 
Along AB two rays ¢ and tf coincide. Also the straight lines 
through A to the points of 6, and through ZB to the points of a are 
double rays of the involution (¢, 2’). The rest of the double rays 
form the bilinear congruence which has a and 5 as directrices. 
2. Let ¢. be a straight line in @; each of its points can be con- 
sidered as its passage P, while its passage Q lies on the straight 
line c= af. If Cz is the point that in [@] corresponds to C=Q 
and ¢', the image of ¢, in [«l, the involution (¢, 2’) associates to t, 
all the rays ¢ of the plain pencil which has C's as vertex and lies 
in the plane (Cat). All the rays tz are therefore singular. 
When tf, revolves round C, ¢, describes a plane pencil round the 
point Cx which in the homology [¢] corresponds to C. The plane 
pencils (¢’) corresponding to ¢, belong to the sheaf [Cs]; their planes 
pass through the straight line C,Cs. 
When C describes the straight line c, Cz describes the straight 
line ez, which in [2] is associated to c. Hence to the singular rays 
t, are associated the rays / of the aval linear complex ez which 
has c; as a directrix. 
