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Some relations connecting the freedoms of manifolds

Published online by Cambridge University Press:  24 October 2008

T. G. Room
T. G. Room
Affiliation:
St John's College
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Extract

The word “freedom” is used in this paper with the same meaning as “Konstantenzahl"—the number of free constants—and corresponds very roughly to the term “degrees of freedom” as used in elementary mechanics. “Freedom of a manifold” is used as a convenient abbreviation for “freedom of the class of manifolds of the same specification as the given one”. At the outset the idea is rather vague; in the background there is an intuitive notion of a definite number associated with any given manifold—the number of conditions that manifolds of the same specification can be made to satisfy. This paper is intented to give some precision to this idea, and to formulate rules which will enable the freedom of a given manifold to be calculated.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 27 , Issue 4 , October 1931 , pp. 518 - 537
Copyright
Copyright © Cambridge Philosophical Society 1931

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References

* 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 γi is a homogeneous quadratic function of (x, y, z). The curve is projected from the line TU into the quartic γ1 γ3 = γ22.