758 DE. PLUCKEE ON A NEW GEOMETEY OF SPACE. 
diameters either intersect the hyperboloid or do not meet it. In the intermediate case, 
where both congruencies are congruent, the corresponding diameter falls within the 
asymptotic cone of the surface. 
65. Conversely, starting from the hyperboloid and any three of its diameters, we may 
revert to the three corresponding congruencies and the series of complexes by means of 
which these congruencies are determined. If especially the three diameters are the 
axes of the hyperboloid, the axes of the three congruencies meet in the same point, the 
centre of the surface, and are directed along its axes. 
There is a double way of reverting from a given hyperboloid to the congruencies, and 
further on to the complexes. The right lines constituting each of its two generations 
may be considered as its rays, while the right lines of its other generation will be found 
to be the directrices of the congruencies passing through the surface. 
66. It might be desirable to support in the analytical way the geometrical results 
explained in the last numbers. For that purpose we may select in order to determine 
the configuration, three complexes of that peculiar description where all rays meet the 
axis. Accordingly the axes of the three complexes O, O', Q" are three of the six direc- 
trices, P, Q, E for instance, confined within the planes j?, q, r. In assuming these 
planes as planes of coordinates XY, XZ, YZ, the three complexes, constituting the con- 
figuration, are represented by equations of the following form, 
O =C +Do- -f-Eg>=0, j 
Q' <; =B's +DV +F (sq — r<r)=0, l (90) 
O" =A"r+Wq + F"(s§ - r<r) = 0. j 
In order to represent by means of a single equation between x, y, z a configuration 
determined by means of three equations between ray-coordinates, these coordinates are 
to be eliminated by means of the following two equations, 
x=rz-\-q, 
y=sz+o , 
to which the third derived one, 
sx—ry=sq—n t, 
may be added. In our case we may at first eliminate sg— r<r, whence 
(B' + Fx')s-F'yr +D'<r = 0, 
(A" - F"y)r + Y"xs + E"§ = 0 , 
and after that § and <7, 
E zr + D;ss = C + Dy + Ear, 
(B' +Far -D'z)s-¥'yr +D'y =0, 
(A" - F"y — Wz)r + F'^s + E"^= 0 . 
