Two conics may be circum‐and‐inscribable to an n-gon; viz. the conies may be such that there exists a singly infinite series of n-gons each inscribed in the first and circumscribed about the second of the conies. In particular they may be circum-andinscribable 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 conies circum-and-inscribable to a triangle (it is presumed there are two or more such planes).

Let the surfaces be Θ, Θ', and let the line Sa tangent to Θ' meet Θ in the points A, B; if through Swe draw two planes as above, then in the first plane the tangents from A, B to the section of Θ' will meet in a point C of Θ; and in the second plane the tangents from A, B to the section of Θ' will meet in a point D of Θ. The points C, D being thus determined the lines AB, AC, BC, AD, BD all touch the surface Θ′, and it is presumed that the surfaces Θ, Θ′ may be such that CD also touches the surface Θ′; viz.