Introduction
In a paper which appeared in 1901, Veneroni cited that in a space Sn of n dimensions, the primals Vnn-1 of order n, which pass through (n+1) general linear spaces Sn-2 lying in Sn form a homaloidal system. Such a transformation is referred to as the Veneroni transformation. Some properties for Sn are given by Veneroni and Eisesland. More specific detail about the transformation in S4 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 S3 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 S4 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 S4 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.