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Published online by Cambridge University Press: 24 October 2008
Given four lines in a plane, we know that the circumcircles of the four triangles formed by them meet in a point. Kühne's theorem states the analogous property for Euclidean hyperspace, viz., that if from n + 2 primes in space of n dimensions we obtain n + 2 simplexes by omitting each prime in turn, the hyperspheres circumscribing the simplexes meet in a point when n is even, but not when n is odd. Mr Grace, in a recent reference to the property, gives a short and interesting proof by help of the rational curves which osculate the given primes and the prime at infinity. It is the purpose of the following note to obtain the result by considering the points of intersection of certain cubic loci.
* “Extension of a set of theorems in circle geometry.” Proc. Camb. Phil. Soc. Vol. XXIV, p. 15 (1928).Google ScholarI ought to mention that the property was communicated to me about four years ago by Mr R. C. Chevalier, of St John's College, Cambridge, who derived it from the alternate vanishing and non-vanishing of skew-symmetric determinants.Google Scholar
* It is understood, of course, that the five planes are such that no four have a common point.Google Scholar
* Grace, . “Circles, Spheres and Linear Complexes.” Trans. Camb. Phil. Soc. Vol. XVI, p. 153 (1898).Google Scholar