1372 
wa, Sl + p(s—p) v} #9, 
y=yntl+q6—a) est, 
22232704. 
The direction of the generatrices of this developable being constant, 
D’ is an enveloping cylinder, 
For 4, = —p:p, and 4,=—q:4q, we obtain analogous results. 
So the theorems hold: 
I. The plane sections of Ov, by planes through any edge of the 
tetrahedron of coordinates are conjugated to those by planes containing 
the opposite edge. 
ll. The developable circumscribed to Occ, along a plane curve, 
the plane of which contains an edge of the tetrahedron of coordinates 
is a cone the vertex of which lies on the opposite edge. 
Il. Any of these enveloping cones cuts Occ, according to curves 
of the system CQ) to which belongs the curve of contact. 
§ 9. Let A(a,6,c) be an arbitrary point. Then the curve of contact 
of the enveloping cone of OO, with A as vertex lies on Occ, itself, 
the equation of which surface is 
yPk y Qk 2Sk — B, 
and on the first polar surface of A with respect to Ove, With the 
equation 
PaxPk-) y Qk 28k 4 Qba PkyQk—-1 28k 4. Sea Pky Qu est 
(BEOS 5 = 0- 
By eliminating B between these two equations we find : 
Payz + Qbuz + Sexy — (P+Q+4+S8)ayz=0. « « (23) 
So the curve of contact always lies on a cubic surface Oi repre- 
sented by (2°). The equation (23) of Oi being independent of 5, 
this surface De is the same for all the surfaces Occ,; so we have 
theorem : 
The locus of the curves of contact of all the surfaces Occ, with 
the enveloping cones with common vertex A is a cubic surface Oh. 
The tangential planes of 0, being at the same time the osculating 
planes of the systems C(p, g, 8) and OC Ne Urs the surface OF is 
also the locus of the points P for which the osculating planes to 
Cp(p.qs) and to Cp(p,, qi, 5) pass through A; this can easily be 
proved directly by making use of the equations (4). 
The surface Oy containing the six edges of the tetrahedron of 
coordinates, four of which also lie on Oe the intersection of Ore. 
