On Four Intersecting Spheres

The interesting results 'On Three Intersecting Circles' obtained by Prof. N.A. Court[4] led me to the present paper. The following results of some interest have been arrived at:

The radical tetrahedron of four intersecting spheres coincides with a diagonal tetrahedron of the desmic system of intersection of those spheres. The pairs of opposite vertices of the system referred to this tetrahedron form pairs of conjugate points for the orthogonal sphere of the given four spheres and are the centres of similitude of the tetrad of associated spheres.

The planes of perspectivity of the eight pairs of complementary tetrahedra of intersection of four intersecting spheres form two tetrahedra desmic with their radical tetrahedron, and are the radical planes of the corresponding pairs of complementary spheres of intersection, and are the planes of similitude of the associated tetrad of spheres.

The diagonal tetrahedra of a desmic system of intersection of four intersecting spheres form the other desmic system of intersection of those spheres.

The eight centres of similitude (other titan the orthogonal centre of four intersecting spheres) of the eight pairs of complementary spheres of intersection of the four given spheres form two tetrahedra desmic with their central tetrahedron and thus form a set of eight associated points.

The desmic system of intersection of four intersecting spheres is inscribed in the one conjugate to that of centres for those spheres and reciprocally their other desmic system of intersection is circumscribed to that of centres for them.

The perpendicular from the orthogonal centre of four intersecting spheres upon the Newtonian plane of their associated tetrad of spheres passes through the circumcentre of their radical tetrahedron.

Each sphere of anti-similitude of the associated tetrad of spheres is orthogonal to eight of the spheres of intersection and to two of the four given intersecting spheres.