1160 
n° + 1.7 + 1,” = 0, also represented by 2,* +- z,* FA == 0. S0 we 
find once more that the plane of o’ is the principal plane and that 
the common enveloping cone of the surfaces ®* touches all the sin- 
gular null-planes. 
By replacing o? by the imaginary circle common to all the spheres, 
we find the metric null-system in which any plane has for null- 
point the foot of the normal out of the fixed point 7. 
We also find a trilinear null-system in the following way. Let 6? be 
any conic and 7’ any point. We then consider as null-plane of any 
variable point Y the polar plane of 7’Y with respect to the cone 
with Y as vertex and 9° as directrix. 
By assuming O, in 7’ and representing 6? by 
a af ae? =, ="), 
we find for the null-plane of X the equation 
Va (voe, + Y2%, + Ut) = (y,’ Ys Fes) te 
So the coordinates 2 of this plane satisfy 
NI = Me Yas = Ms YI =H Ya" + Ys’) 
As these relations are identical to those of (3) this null-system is 
equal to the former. 
§ 7. We now pass to bilinear null-systems where y = 2. 
Then the locus of the null-points of the planes of a pencil with 
axis lis a twisted cubic curve (l)* cutting / twice. 
Analogously the null-planes of the points of a line / envelope a 
developable with index 3 (torse of the third class), i.e. they osculate 
a twisted cubic. 
The locus of the null-points of the planes passing through a point 
P is a cubic surface (P)'. 
Two surfaces (P)* and (Q)* have the curve (/)* determined by the 
line L= PQ in common. [n general they admit as completing inter- 
section a twisted sextic o°, cutting (/)* in eight points and forming 
the locus of the singular null-points, each of which bears a pencil 
of null-planes. (If these planes were to envelope a cone o° has to 
be manifold curve on (P)* and this is impossible if we surmise that 
the intersection of (P,* and (Q)* breaks up into two parts only). 
The axes s* of the pencils of null-planes through the points S of 
6° form a scroll of order eight; for the points of intersection of 0’ 
and (/)* determine eight null-planes through /, each of which has a 
point S as null-point and contains therefore an axis s*. 
The surfaces (P)*, (Q)* and (R)* have the singular curve 6° in 
common and moreover one point only, the null-point of the plane 
