550 
[550 
550. 
PROBLEM AND HYPOTHETICAL THEOREMS IN REGARD TO 
TWO QUADRIC SURFACES. 
[From the Messenger of Mathematics, vol. n. (1873), p. 137.] 
Two conics may be circum-and-inscribable to an w-gon; viz. the conics may be 
such that there exists a singly infinite series of ?i-gons each inscribed in the first and 
circumscribed about the second of the conics. In particular they may be circum-and- 
inscribable to a triangle. 
The following problem arises: 
Consider two given quadric surfaces and a given line S; to find the planes through 
S, which cut the surfaces in two conics circum-and-inscribable to a triangle (it is 
presumed there are two or more such planes). 
Let the surfaces be ©, ©', and let the line S a tangent to ©' meet © in the 
points A, B\ if through S we draw two planes as above, then in the first plane the 
tangents from A, B to the section of ©' will meet in a point G of ©; and in the 
second plane the tangents from A, B to the section of ©' will meet in a point D 
of ©. The points G, D being thus determined the lines AB, AG, BG, AD, BD all 
touch the surface ©', and it is presumed that the surfaces ©, ©' may be such that 
GD also touches the surface ©'; viz. in this case we have a tetrahedron ABGD, the - 
summits of which lie in the surface ©, and the edges touch the surface ©'; and not 
only so, but it is further presumed that the surfaces may be such that starting from 
any point A of © and using either any tangent or a properly selected tangent AB 
of ©', it shall be possible to complete the figure as above; or, what is the same thing, 
the surfaces may be such that there exists a doubly or a triply infinite series of 
tetrahedra, the summits of each lying in © and its edges touching ©'. It is also 
presumed that the faces of the tetrahedra all touch one and the same quadric sur 
face ©".