589 
ciated to the generatrices of the cone A*; it happens jive times that 
such a plane passes through the corresponding straight line, so that 
this line is a bisecant / of the seeond kind passing through P. 
Hence the order of the congruence formed by these bisecants is five. 
To each ray / of one of the congruences (1,3) and (5, 3) corres- 
ponds a plane pencil of straight lines / which project a line d 
from a point of the corresponding conic £*. For the lines / of the 
second kind this point coincides with D’, so that the congruence 
(5,3) is transformed into itself; for those of the first kind it is an 
arbitrary point of the conic £*. 
A plane V intersects the conics 4? in the points of a curve cf 
that has a node in the intersection of the plane V ‘with the line 
A A,, and the lines d in the points of a conic c°. Between the 
points of the curves c* and c’ there evidently exists a correspondence 
(1, 2). The three points of intersection of these curves lying outside 
the intersections of the plane V with the lines a,, a, and a, and 
with the two transversels 6, and 6, of the four lines 4,A,, a, a, 
and a,, are points D, hence coincidences of this correspondence. 
The lines connecting associated points of this correspondence, in 
other words the rays /' lying in the plane JV’, envelop therefore a 
curve of class five. 
The rays U corresponding to the rays | of the congruence (1,3) 
form consequently a line complea of order five. 
The degenerate curves 0° of the second series, found in § 3, do 
not contain any singular points. 
§ 12. Coincidences. A line A produces a coincidence if its sup- 
porting points P and Q coincide with their associated points 1’ 
and Q’. 
The involution (P, P’) has a finite number of coincidences, viz. the 
points A,, A, and the six points D found in §5, in which the trans- 
versals 4,,, ete. cut the corresponding planes a,, etc. The line A,A, 
and the lines connecting the points A, and A, with the points D 
are therefore rays of coincidence. 
Let us further consider a line / through the intersection D, of 
the line 4,,, with the plane «,,. This line is bisecant of a degene- 
rate y® formed by the line b,,, and a conic £* in the plane «,,; in 
the point D, this conic touches the plane brought through the lines Land 
be 
of which the supporting point P lies on the line 6,,,, the supporting 
point © on the eonie k?, to approach D,, we get such a straight line /. 
The point P/ associated to P lies on the line 6,,, and is barmoni- 
39 
For if we cause the two supporting points of a bisecant PQ 
Proceedings Royal Acad. Amsterdam. Vol. XXII. 
