### Properties of the Veneroni Transformation in S_{4}

American Journal of Mathematics, Vol. 58, No. 3 (July 1937), 639-645

Introduction

In a paper which appeared in 1901, Veneroni cited that in a space S_{n} of n dimensions, the primals V^{n}_{n-1} of order n, which pass through (n+1) general linear spaces S_{n-2} lying in S_{n} form a homaloidal system. Such a transformation is referred to as the *Veneroni* transformation. Some properties for S_{n} are given by Veneroni and Eisesland. More specific detail about the transformation in S_{4} is given by J. A. Todd and by Virgil Snyder. The bilinear equations defining the transformation have not heretofore been published, however. They are derived in this paper and further properties are investigated with their aid. Emphasis is laid on the study of involutorial Veneroni transformations. In S_{3} any Veneroni transformations can be made involutorial by a proper choice of the frame of reference—an elegant derivation is given by H. F. Baker. We show in this paper that in S_{4} this is no longer true; one condition among the coefficients of the equations becomes necessary for an involution. If is found that quite generally, but not always, the involutorial case can be represented as a polarity with respect to four composite quadric primals, and the fundamental elements are considerably more specialized than in the more general involutorial transformations studied by Schoute and by Alderton. Furthermore, *there exist Veneroni transformations in S*_{4} which are involutorial; but the bilinear forms cannot be represented as polarities with respect to quadric primals by any linear transformations. Properties which have already been found by other investigators will be included for the sake of clarity and completeness, when necessary.