Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-28T09:10:45.089Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.