* Cf. Grace, , Proc. Camb. Phil. Soc. 25 (1929), 424CrossRefGoogle Scholar. The impulse to undertake this investigation is largely due to Mr Grace's paper.

* For references and fuller description see § 1·1, p. 523.

* In detail the process is this: take the line determined by the two points (a ₀, a ₁, a ₂, 0), and (β₀, β₁, 0, β₃); the condition that every point (a) + λ (β) should lie on the surface is then expressed by five equations involving the coefficients of the equation of the surface and the a and β. When these are satisfied the surface contains the definite line; the condition that the surface should contain “some” line is obtained by eliminating the a and β (in all four independent quantities) from the five equations.

* See e.g. Segre, , Ann. di Mat. 22 (1894), 47Google Scholar, where equations I and III below are given.

* Cf. section 2, example (r).

* For further explanation see § 1·3.

* T is such that every point of Θ corresponds to at least a line of it (exceptionally, if the point is singular on every Φ, it corresponds to some more complicated locus than a line).

* This is not a formula which necessarily enables us to calculate f _U, since probably no more is known about _[m]f _Θ and _Uf _T than about f _U. It can however be used in many cases—cf. examples (d), (e), (f) at the end of the section—and it leads on to the results of section 2.

† The contribution from the other source is still zero, since the functions ø are general.

* Putting i _U for the number of independent absolute invariants of U, and t _U for the freedom of collineations of U into itself (usually one or other of these quantities is zero), we can write this equation as

cf. Grace, loc. cit.

† We take here only one of the types of normal rational scroll; the process applies equally well to scrolls with directrix curves of lower order.

* We are not yet dealing with the case where a point of projection lies on V (and therefore on U). Then Ψ has to pass through an additional fixed point of the [m] and N _ψ is reduced, but account is taken of this in another way.

* In this case U and V are normal in the same space.

† It is to be remembered that the projections of U and of U ^(1,−1) must agree not only in their geometric specification, but also in the type of parametric equations by which they are represented on [m]; it seems probable in fact that this second cause cannot arise.

* Mr Babbage suggested this example.

† This example was suggested by Professor Semple.

* This problem was propounded and solved by Mr White. The equations of the particular take the form t ² = γ₁, u ² = γ₃, tu = γ₂, where γ_i is a homogeneous quadratic function of (x, y, z). The curve is projected from the line TU into the quartic γ₁ γ₃ = γ2².