and Or breaks up into the curve of contact and these four edges. 
The tangential plane of O4 in any point of one of these edges is 
the same for all the points of this edge and different from the faces 
of the tetrahedron of coordinates. As we always have S<— Q 
and we suppose provisionally that P > — Q, the tangential planes 
of O.-, along the four edges coincide with faces of the tetrahedron 
of coordinates. So each of the four edges belongs to the intersection 
a number of times indicated by its multiplicity on O,,,. 
Now the edge OX, is always S4-fold on O,., and YZ, is always 
(S+Q+P)k-fold, while for P>>—Q the edge X, VY, is Sk-fold 
and the edge OZ, is — Qk-fold. So the four edges represent together 
(3S-+-P)k common right lines. The total intersection of O4 and Tan 
being of the order 3(P?+S)k, there remains a curve of contact of 
order 2Pk. 
In the case P< — Q the edge X, Y,, counts (P+Q+S)é times 
on O,,. and the edge OZ, counts Pk times. Then the four edges 
represent (8S+3P+2Q)s commen right lines belonging to the 
intersection and therefore the curve of contact is of order — 2 Qk. 
For P=— Q which implies S=S+ P+ Q the tangential plane 
of O... along OZ, is no more constant and therefore this plane does 
not coimeide with the tangential plane of O4 along this edge which 
is constant; likewise for the edge Y, Z,. So the multiplicity of 
these edges as parts of the intersection still remains equal to their 
multiplicity on 0. 
Now the edge X, Y, is Sk-fold on Oj, and the edge OZ, is 
Pk-fold. The order of the curve of contact is 2P = — 2Qk. 
From P=— Q we deduce 
rn hd 
ies client NE ter Pr OF iy Sigs lw the: first ease Or,’ ist ar ruled 
surface (see § 3, § 14), in the second a plane (see $ 3). 
As in general the point A does not lie on the surface O,,, it 
neither lies on the curve of contact and the order of the enveloping 
cone to QO... with vertex A is equal to the order of the curve of 
contact. So we find the theorem: 
The order of the enveloping cone to Ove, with an arbitrary vertex 
A is the larger of the two numbers 2Pk and — 2Qk. 
If A lies in one of the faces of the tetrahedron of coordinates, 
0% breaks up into the plane of that face and into a quadratic cone the 
vertex of which coincides with the opposite vertex of the tetrahedron. 
If A lies on one of the edges of the tetrahedron of coordinates, 
