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The Genus of a Developable Surface

Published online by Cambridge University Press:  24 October 2008

H. F. Baker
H. F. Baker*
Affiliation:
St John's College
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Extract

Cayley's remark that the formula by which the genus of a surface, according to Clebsch's definition, may presumably be computed leads to a negative number in the case of a cone, or a developable surface, or a ruled surface in general, has great importance in the history of the theory. But it would appear, from various indications, that, for a developable surface at least, it is more often quoted than read. I have thought therefore that the following simplifying remarks may have a use. Cayley uses formulae, due to Salmon and Cremona, without reference to the memoir where these are given in detail. Of two of these, for the number of tangents of a curve which meet it again, and for the number of triple points of the nodal curve, proofs by the theory of correspondence are extant; for the present purpose it is only necessary to have the sum of these two numbers. I do not know whether it has been remarked that there exists a remarkable formula for this sum, very similar to, and including the ordinary formula for the number of triple points of a general ruled surface (and like this probably capable of a direct proof by the theory of correspondence). For the genus of the nodal curve, deduced by Cayley from the Salmon-Cremona formulae, a proof by the theory of correspondence (in the general case, sufficient for the purpose in hand, in which i = τ = δ = δ′ = 0) is added here, which seems to have a certain interest.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 31 , Issue 2 , April 1935 , pp. 156 - 158
Copyright
Copyright © Cambridge Philosophical Society 1935

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References

* Cayley, , Math. Ann. 3 (1871), 526–9Google Scholar; Collected Papers, 8, 394–7.Google Scholar

† Cayley, , Quarterly Journal, 11 (1871), 294–317 Google Scholar; Collected Papers, 8, 72–91.Google Scholar

‡ Cayley, , Collected Papers, 8, 395.Google Scholar

§ Cayley, , Phil. Trans. 159 (1869), 201–29 (411)Google Scholar; Collected Papers, 6, 356.Google Scholar

* Cayley, , Collected Papers, 8, 85.Google Scholar

† In Cayley's, Collected Papers, 8, 395 Google Scholar, it is stated that the (β) points are stationary on b; and the (γ) points stationary on c; and this misprint is copied in Salmon, , Geometry of three dimensions (1882), 558 Google Scholar, and reproduced in the new edition, Salmon-Rogers, 2 (1915), 268.

‡ Cayley, , Collected Papers, 8, 86.Google Scholar

§ Noether, , Acta Math. 8 (1886), 161–92 (182).Google Scholar

∥ Cayley, , Collected Papers, 8, 76 Google Scholar; or from the formulae given in Baker, H. F., Principles of Geometry, 6, 38, 32 Google Scholar, where ξ is used for γ.

* Baker, H. F., Principles of Geometry, 6, 11 Google Scholar; the number 2 is twice omitted in line 6 from the foot.