489 
nience sake here referred to as the ‘correlating surface’, which has 
conical points at the points at infinity of the axes of coordinates and 
intersects the plane at infinity along the lines at infinity of the three 
planes of coordinates. It follows from this that two such surfaces 
have a twisted sextie 4° in common, which has three double-points, 
one at the point at infinity of each axis. Three surfaces therefore 
have six points in common with finite coordinates. 
We conclude from this that the correspondences 7'p, T'q and Tr 
have six triplets of rays in common. One of these consists of the 
lines 6, c and d which pass through the point of intersection A* of 
3, y and 0. The other triplets furnish each a ruled surface passing 
through P, Q and R. Hence through any three points there pass five 
surfaces of the system. 
Moreover, if it is taken into consideration, that by the rays of a 
plane too a trilinear correspondence is established between the pencils 
(6), (ec) and (d), it follows that there are five surfaces which are 
tangent to three given planes, six passing through two given points 
and tangent to a given plane, and six which pass through a given 
point and touch two given planes. 
2. If the curve of intersection £° of two “correlating surfaces” 
is projected on a plane of coordinates, the result is a plane curve 
2* with double points at the points at infinity of the two axes of 
coordinates. It follows from this that the surfaces (bed), which pass 
through two given points P and Q, determine a (2,2)-correspondence 
between the pencils (4) and (c), obtained by conjugating the line 
6 of such a surface to its line c. The same holds for the surfaces which 
pass through two infinitely near points and thus are tangent to a 
line / at a point S. If we project (6) and (c) from S we obtain two 
pencils of planes between which a (2,2)-correspondence exists. In 
this correspondence the plane which contains the axes SB and SC 
of these two pencils, and which belongs to both pencils, is conju- 
gated to itself. For, in fact, this plane intersects B and y along two 
lines 4 and ce which meet at a point 7’ lying on the line of inter- 
section of @ and y. If to these lines we conjugate the ray of (d) which 
passes through the point of intersection of ST'and d, the correspond- 
ing (bed) breaks up into two planes, the line of intersection of 
which contains S. The here considered lines 4 and c are therefore 
generators of a ruled surface (hed), which is at S tangent to /, 
from which it follows indeed, that the plane passing through SB 
and SC is conjugated to itself in the (2,2)-correspondence. Each pair 
of planes of this correspondence furnishes by its line of intersection 
a generator of a ruled surface which is tangent to / at S. Hence 
