It is well known that the canonical system of curves on an algebraic surface is only relatively invariant under birational transformations of the surface. That is, if we have a birational transformation T between two surfaces F and F′, and if K and K′ denote curves of the unreduced canonical systems on F and F′, then

where E and E′ denote the sets of curves, on F and F′ respectively which are transformed into the neighbourhoods of simple points on the other surface.

We investigate in this paper a certain special family of quadric varieties, that is of in [R]. Now among the more important properties of a quadric in [R] is that it possesses a system or systems of “ generators,” i.e., the quadric may be taken as the locus of certain families of subspaces, the behaviour of these depending on the parity of R. If R is even, a quadric V2n–1 in [2n] contains a single family of [n—l]'s, so it seems likely that in discussing special families of quadrics in [2n] an important type will be obtained by constraining the quadric to pass through a number of [n — l]'s.

It is proposed here to consider the sequence un determined by the relation

where, in particular,

and initially u1 = θ1. The following is the main result to be proved.

The circumstances explained in the footnote on p. 61 of the former paper with this title might well have necessitated a re-writing of the whole. Fortunately it appears that only a few short comments are required. For example it may be noted that the first question in §4 can be answered by counting the degrees of freedom in the two configurations. Eight points of a twisted cubic have freedom 20; four pairs of planes drawn at random through four lines of a regulus have freedom 21; therefore the eight planes of the second paragraph of §4 cannot always lead back to the eight points of §2. This is corroborated by the corresponding numbers in [4], which are 31 and 34.

In Kottler's theoretical discussion1 of the diffraction of a plane wave of monochromatic light of wave-length 2π/k by a black halfplane, the function

where (r, θ, z) are cylindrical coordinates, plays an important part. In particular it is necessary to have asymptotic formulae for f (r, θ), valid when r is either very large or very small compared with the wave-length.