In the first section of this paper we illustrate the use that can be made of higher space in dealing with the problem of resolving a given Cremona transformation into the product of simpler Cremona transformations. In the second section we restrict ourselves to a particular large but finite class of Cremona transformations of [3], those of genus one, and show that these can all be built up from the four following simple types:
(1) The bilinear transformation T3,3, determined by three equations bilinear in the coordinates of the two corresponding spaces; in the most general case of this both the direct and the reverse homaloidal systems consist of cubic surfaces passing through a non-degenerate sextic of genus three;
(2) Three transformations Tn, n (n = 2, 3, 4) in which the homaloidal surfaces may in each case be obtained by taking in [4] a primal V of order n which has two (n− l)ple points, and projecting on to a given [3] from one of these points the sections of V by primes through the other; for n = 2 we have the familiar quadroquadric transformation determined by quadrics through a conic and a point.