483 
A,A,, A,A,, A,A,, 4,4, and A,A,, A,A,, A,A,,A,A, are taken for 
basal edges, then the curves «* having a line a, drawn throngh A, 
for a chord will all lie on the cone a,*, which is determined by a 
At each of the two points of intersection of a, and the curve 8,° 
which has a, for a chord, three rays # meet which, in two differents 
groups of the J‘, are conjugated to t=a,. It follows from 
this that: 
Each singular ray passing through one of the ten principal points 
(Ax, By) is by the transformation (t, t’) converted into a ruled surface 
of the sixth deyree. 
The curve #,°, which has a, for a chord, has also a chord a, 
passing through A,; the latter meets the cone a,? at A, and at a 
point P. Through P passes a curve « having with 3,° the chords 
a, and a, in common; hence a, lies on the ruled surface a,°, corres- 
ponding to a,. 
The plane Ba, intersects the cone a,’ also along an edge a,'. 
On a, and a,’ the curves a? of this cone determine two projective 
point-ranges; hence through B, pass two chords of curves «° lying 
on a,*. The chords of the curves «* meeting a, therefore constitute 
a quadratic complex. 
The cone of the complex at the point Bj, has four edges in common 
with the cone which projects 3,° from Bz. Hence the ruled surface 
a,° has a quadruple point at each of the principal points Bz. 
The line a,,—A,A, is a principal ray of the /'*. For the curve 
8, which has a,, for one of its chords, determines a group of the 
7” with each of the curves «*; a,, therefore belongs to oo? groups 
and the rays ¢ which are conjugated to a,, constitute a congruence. 
The chord ¢ of 8,,* which passes through a point P, is at the same 
time chord to an a’ 
P lie three chords ¢’ of 8,,*, each of which is at the same time 
chord to a curve a°. Hence the ray t=a,, is conjugated to the 
rays t of a congruence (1,3). 
3. The ray A,4, is a common chord to oo? pairs a’, 3* and is 
therefore a princinal ray of [® and conjugated to the rays ¢' of a 
and therefore conjugated to {— a, In a plane 
line-congruence [1] 
Each point S of A,B, carries one a’ and one 3’, and consequently 
three rays ¢'; hence the order of [t'| is three. 
The curves «’ which have A,4, among their chords lie on a 
cone a,* and determine an /* on the conic which a,* has in common 
with a plane ® The chords of the «* which lie in ® therefore 
envelop another conic. Similarly ® contains also a conie which is 
3 
enveloped by the chords of the curves p? which have A,B, among 
