594 
PROBLEMS AND SOLUTIONS. 
[485 
through the line xx all pass through the same two points, and since B, G are two 
of these conics, B and C meet xx in the same two points X, X'; similarly G and 
A meet yy' in the same two points Y, Y'; and A, B meet zz' in the same two 
points Z, Z'; that is, we have the conics A, B, C intersecting 
B, G in the two points X, X', 
G, A „ „ Y, T, 
A, B „ „ 
hence taking on the conics A, B, G the points a, (3, 7 respectively, and drawing a 
quadric surface 2 through the nine points X, X', Y, Y', Z, Z', cl, /3, 7, this meets the 
conic A in the five points Y, Y’, Z, Z', a; that is, it passes through the conic A, 
and similarly it passes through the conic B, and through the conic C. 
Consider how any plane whatever through 0 intersecting the conics A, B, C in the 
points L and L', M and M', X and N' respectively; the section of the quadric surface 
2 by the plane in question is a conic through the six points L, L', M, M', X, N'. 
But the section of the surface includes a conic through these same six points, and which 
is consequently the same conic; in fact, the section of the surface by the plane in 
question includes a conic, and since every section through the line LL' includes a 
conic through the same two points, and one of these conics is the conic A which 
passes through the points L and L', the conic in question passes through the points 
L and L'; and similarly it passes through the points M and M', and through the 
points X and X'. That is, for any plane whatever through 0, the section of the 
surface includes the conic which is the section of the quadric surface 2, and the 
surface thus includes as part of itself the quartic surface 2. 
The foregoing demonstration ceases, however, to be applicable if 0 is a point on 
the surface, and the conic included in the section through 0 is always a conic passing 
through the point 0. In the case where 0 is a non-singular point of the surface 
(that is, where there is at 0 a unique tangent plane) a like demonstration applies. 
Take through 0 a section, and let this include the conic A ; on i take any point 
0' and through 00' a section including the conic B ; we have thus the conics A, B 
intersecting in the points 0, O'. Take through 0 any plane meeting the conics A, B 
in the points X, Y respectively—the section by this plane includes a conic G passing 
through the points 0, X, Y; and each of the conics A, B, C touches at 0 the same 
plane, viz. the tangent plane of the surface. Hence, taking on the conic A the point a, 
consecutive to 0, and any other point cl; on the conic B the point /3, consecutive 
to 0, and any other point (3'; and on the conic G a point 7'; we may, through the 
nine points 0, cl, ¡3, O', a, (3', X, Y, 7' describe a quadric surface 2; this will touch 
at 0 the tangent plane of the surface, that is, it will touch the conic 0, or (what 
is the same thing) pass through a point 7 of this conic consecutive to 0. Hence the 
quadric surface meets the conic A in the five points 0, O', cl, oí, X, that is, it entirely 
contains the conic A ; similarly it meets the conic B in five points 0, O', B, B', Y, 
that is, it entirely contains the conic B; and it meets the conic G in the five points 
0, 7, X, Y, 7', that is, it entirely contains this conic. And it may then be shown as