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their chords. Hence in ® lie four chords t' conjugated to A,B,, so 
that the class of [t'] is four. 
Through A, there passes a curve 8; the quadrie cone projecting 
this curve from A, is intersected by the projecting cones of the 
curves a@ which have 4,B, among their chords along sets of three 
edges ¢’. It follows from this that the congruence [t’] has singular 
points of the second order at A, and B,. 
Every 8° of which A,B, is a chord has a chord a, passing through 
A,; the latter chord intersects the cone a,* determined by A,B, at 
a point P (in addition to A,). The a’ passing through P has the 
chord a, in common with @°. It appears from this that A, is a 
singular point of the congruence (3, 4). ; 
A plane ® through A, intersects the quadratic cone 6,? determined 
by A,B, along a conic, on which the curves 8° cut out an invo- 
lution Z*. Hence through A, pass two of the chords in the plane 
P belonging to the curves p’. Thus we find that the principal points 
Ar and B (k#1) are singular points of the second order with 
respect to the congruence (3, 4) conjugated to A,B,. 
The line A,B, is a chord of the figure « composed of a,, = A,A, 
and an a°. The cone 8,’ which contains the curves 8° of which 
A,B, is achord determines an involution /* in the plane «,,, = A,4,A,, 
the sets of which involution consist of the transits of the curves 6°. 
The chords ¢’ of the congruence [t’] which lie in «, therefore 
envelop a conic. It follows from this that the planes az, and Brin 
are singular planes of the second order with respect to the con- 
gruence (3,4) conjugated to A,B,. 
4. Every ray t in the plane a@,,, is singular since it is a chord 
to all conics a? lying in «,,,. Together with the curve 8% which 
has ¢ among its chords, the oo’ curves a? evidently determine oo! 
groups of rays ¢’; each group consists of the two additional chords 
of 6? which lie in a@,,, and of seven rays ¢’ resting ona,,. All these 
sets of seven belong to the ruled surface (¢’)* engendered by the 
chords of #* which rest on a,,; they constitute an /7 on this surface. 
To the rays [t] of the plane a,,, as a whole are conjugated the 
rays fl’; of the special linear complex which has a,, for directrix: 
These rays form c* groups of seven rays. 
Every ray t= dim Bem is a principal ray of the /*°: for, in fact, 
it is a chord to oo’ conics a? and to oo! conics B*, and therefore 
conjugated to oo* groups of rays 4’. 
Now consider for example the line ¢= «‚ 8,,, as a common chord of 
two definite conics @ and g°. Of the corresponding lines ¢’ two lie in 
a,,,; they join the transit of 6,, with the two transits of 8*. Similarly 
