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Published online by Cambridge University Press: 03 November 2016
The elementary properties of generators of a ruled surface, and the existence of a line of striction when the surface is skew, are well known to readers of this journal. We propose to show that many of these properties do not belong exclusively to ruled surfaces; but that a family of curves on any surface possesses a line of striction and a focal curve or envelope, though these are not always real. When the surface is developable, and the “curves” are the generators of one system, the focal curve is the edge of regression. We shall also see that the properties to be established for a family of curves on a surface are analogous to some of the leading properties of congruences of curves in space, the line of striction corresponding to the surface of striction or orthocentric surface, and the focal curve to the focal surface of the latter.
page 1 note * See a recent paper by the author, On Congruences of Curves, §6. The writing of the present paper has suggested to the author that the term “surface of striction” may be preferable to the term “orthocentric surface” originally employed.
page 2 note * Cf. the author’s Differential Geometry, Art. 25.
page 2 note † This is also the focal curve of the family u=const., and of the family φ(u, v)=const.
page 2 note * More generally, when E is not equal to unity, the line of striction of the parametric curves v=const. is given by
.
page 2 note § All the differential invariants (grad., div., curl, etc.) of this paper are the two-parametric invariants introduced by the author in a recent paper, “On Differential Invariants in Geometry of Surfaces, etc.,” Quarterly Journal of Math., 1925.
page 3 note * Ibid. Art. 8.
page 3 note † All this Is easily verified from the value H2=a2u2+2bu+sin2θ, where a, b, θ are independent of u. See the author’s Differential Geometry, Chap. VII., Art. 70.
page 3 note ‡ Using the expansion formula divφs=φ divs+s. ∇φ, and the elementary results ∇sinθ=cosθ∇θ, etc.
page 3 note * See the author’s Differential Geometry, Art. 72.
page 4 note * On Differential Invariants in Geometry of Surfaces, etc., Art. 11 (36).
page 4 note † Differential Geometry, Art. 49.
page 4 note ‡ Ibid. Arts. 53-55.
page 4 note § On Differential Invariants, etc., Art. 11 (37).
page 5 note * See Differential Geometry, Art. 52.
page 5 note † Differential Geometry, Art. 41.
page 5 note ‡ On Differential Invariants, etc., Art. 4 (7).
page 5 note § Cf. Bianchi, Geometria Differenziale, vol. i. p. 124, § 48.
page 6 note * On Differential Invariants, etc., Art. 4 (6).
page 6 note ‡ Ibid. Art. 7 (23).