( 574 ) 
which passes through the points P,,0,X, ,V,,.2,,,(v=0, ©,—a*,-—b*,—c’), 
and which is thus an orthogonal eubie hyperbola w*. Each of its 
points P, determines an ellipsoid, for which P,P, is the normal 
ins P. 
Through a given point P, pass @' curves w'; their “foci” (Null- 
punkte) P, are indicated by 
aa) (af uti by (bi a) yn, esa, (CN) A ENG) 
so they lie on the normal having P, as footpoint. 
These curves are all situated on the surface determined by the 
equations 
(2 + ua, n= (0° + u)y, (te Wes 
ZES 
Pe 
SS SS ar: ; = (6) 
atv bv ety 
or also by the equation obtained from these by elimination of u 
and v 
| ae) & @ 
En 
| 
© 
NOS de B (7) 
BNA ZZ 
This same equation we obtain out of (3) if we express the 
coordinates of rays in the coordinates of points. So the locus of 
the curves w* passing through P, is the complex cone of P,. 
Corresponding to this we find out of (6) for v= const. a right 
line through P,, whilst w = const. indicates a curve w’ through P,. 
4. Out of the preceding ensues that all bisecants of a curve o* 
are rays of the complex. This is further contirmed by the caleulation 
of the coordinates of rays of the bisecant (v, v'). We find out of (4) 
aay (v' Et 2) 0) etc 
(a? + v)(a* Hv) 
b*c*y 2, (b° —c*) (v! -— v) 
== = —— EEG 
(0? Ho) (C° + v) (b Hv) (0? Hw) 
from which ensues readily 
appa + O’p ps + PaP = 
(i 
5. The planes of coordinates and the plane at infinity are the 
principal planes of the complex. The complex cone of a point lying 
in a principal plane must degenerate. 
We truly find out of (7) for z,—=0 the planes z =O and 
(a? — c*?)y,a@—(b? —c)a,y — (at —D*)ayy,. . « ~ (8) 
In. connection with this the curve w* consists now of the hyperbola 
dl} (at — bay bye ae y—=0 ... . (9) 
