Orthogonal hyperspheres

Umbilical projection (12], 14]) is a process suggested to derive results rather quickly in regard to four intersecting spheres 17] andn+1 intersecting hyperspheres in ann-space 18]. The same has been used with an advantage to deduce a porism on 2n+5 hyperspheres in ann-space 23]. The purpose of this paper is to concentrate on mutually orthogonal hyperspheres only and to illustrate simultaneously once again the utility and facility of this tool to arrive at a number of new and interesting results as follows:The 2(n+1) intersections ofn+1 mutually orthogonal hyperspheres in ann-space, takenn at a time, give rise to 2 ⁿ pairs ofsemi-inverse 22] simplexes, perspective from their radical centreH, such that the 2 ⁿ primes of perspectivity coincide with their 2 ⁿ hyperplanes of similitude and form anS-configuration (S-C) 15] with theircentral simplex S(A) as itsdiagonal simplex. Everysimplex of intersection introduced here isisodynamic 25] such that itstangential simplex, circumscribed to it along circumhypersphere, is perspective to it from itsLemoine point L. ItsLemoine hyperplane l, as the polar prime ofL w. r. t. it, is the same as that of itscomplementary simplex of intersection and coincides whith their prime of perspectivity such that their 2(n+1) altitudes are met by their commonBrocard diameter through their Lemoine points. The 2 ⁿ Brocard diameters of the 2 ⁿ pairs of complementary simplexes of intersection concur atH. The hyperspheres of antisimilitude of the given hyperspheres, having centres in a prime of similitude, form the commonNeuberg hyperspheres of the pair of semi-inverse simplexes, having this prime as their common Lemoine hyperplane, are consequently orthogonal to their cirumhyperspheres whose radical hyperplane, too, coincides whith this prime, and therefore belong to acoaxal net 15] passing through the pair of their commonNeuberg points on their common Brocard diameter. The second centres of similitude of the 2 ⁿ pairs ofcomplementary hyperspheres of intersection form the 2 ⁿ vertices of the dual 15] of the (S-C), whithS(A) as common diagonal simplex, as its polar reciprocal w. r. t. the common orthogonal hypersphere of then+1 hyperspheres, the first centres of similitude coinciding atH.Due inspiration is derived from the works ofCourt (2]–9]) on mutually orthogonal circles and spheres. Presented by G. Hajós