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Published online by Cambridge University Press: 24 October 2008
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.
* The genus of a Cremona transformation of [3] is defined as the genus of a general plane section of a general member of the homaloidal system which determines the transformation; the genus is clearly an invariant of the transformation.
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* If we project the surface on to a plane 
from the osculating [5] at one of its points P, the third neighbourhood of P is represented by a rational cubic Γ with a node O, say. The prime sections of the surface project into a system ∑ of decimics whose base points can be seen without much difficulty to be a sextuple point at O and six triple points on Γ. ∑ is transformable into a system of quartics through six points.
* H. P. Hudson, loc. cit. pp. 382–7.
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* Γ meets C in three points because the plane of C cannot meet a sextic outside C. Therefore the free curve of intersection of a sextic with one of the transforming quadrics is a (3, 3) curve with three double points and hence is elliptic.
* The generators of the tangent cone correspond to the neighbourhoods of the points of the projecting sextic, i.e. they are the projections of the tangent solids to 
at these points.
* The cubics of ∑′ touch the quadric through Γ′ along l′.
* T 4, 4 is of the type previously described which is obtainable from a quartic primal in [4] with two triple points.
* On φ′ there is a pencil of such conics meeting R in eight points; they arise from the neighbourhoods of O on the different quadric cones through the tangents to the branches of Γ at O.
† H. P. Hudson, loc. cit. p. 387.