1. A proof of Dupin's theorem with some simple illustrations of the method employed. 2. Two methods of obtaining Cayley's condition that a family of surfaces may form one of an orthogonal triad. 3. An extension of Dupin's theorem to the case in which a family of surfaces is cut orthogonally by two other families which intersect at a constant angle, with the condition that a family may be capable of being cut in this manner

F. E. Edwardes

doi:10.1017/S0013091500033794

1. A proof of Dupin's theorem with some simple illustrations of the method employed. 2. Two methods of obtaining Cayley's condition that a family of surfaces may form one of an orthogonal triad. 3. An extension of Dupin's theorem to the case in which a family of surfaces is cut orthogonally by two other families which intersect at a constant angle, with the condition that a family may be capable of being cut in this manner

Published online by Cambridge University Press: 20 January 2009

F. E. Edwardes

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A proof of Dupin's theorem with some simple illustrations of the method employed.

Before plunging into Dupin's theorem, I think it well to speak of certain infinitesimal rotations which play a part in the proof. By an infinitesimal angle of the first order is meant an angle subtended at the centre of a circle of finite radius by an arc whose length is an infinitesimal of the first order. If we neglect infinitesimals of the second order, equal infinitesimal rotations of the first order about axes which meet and are separated by a small angle of the first order are identical. For instance, if AB and BC be elements of a curve of continuous curvature, an infinitesimal rotation about AB may, if we prefer it, be regarded as taking place about BC; and again, if OA, OB, OC be a set of rectangular axes, small rotations about OA, OB, OC may be regarded as taking place in any order. For if P be a point on a sphere of finite radius, and PQ, PR be the displacements of P due to equal infinitesimal rotations of the first order about two diameters separated by a small angle of the first order, the angle QPR is the angle of separation of the axes, and it follows that QR is an infinitesimal of the second order. Further, if the radius of the sphere is an infinitesimal of the first order, QR is of the third order of small quantities.

Type: Research Article
Information: Proceedings of the Edinburgh Mathematical Society , Volume 29 , February 1910 , pp. 41 - 64

DOI: https://doi.org/10.1017/S0013091500033794 [Opens in a new window]
Copyright: Copyright © Edinburgh Mathematical Society 1910

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* It is important that I should be quite clear as to the convention I adopt with regard to positive and negative rotations. I do not care whether the axes of coordinates be right or left handed: I only stipulate that a positive rotation through a right angle about the axis of x shall bring the axi. of y into the position formerly occupied by the axis of z, and so on with cyclic interchanges of the letters in the order x→y→z→x. When I use axes O(A, B, C) or O(1, 2, 3) I suppose them drawn so that they can be made to coincide with O(x, y, z).

* M. Fouché has anticipated the idea which is the basis of my proof in a paper entitled “Démonstration géométrique du théorème de Dupin,” Nouvelles Annales tie Mathématiques, ser. 4, t. 7 (1907). M. Fouché denotes the geodesic torsions etc., by τ, τ₁, τ₂, and having obtained an OA expression for τ, observes that the geodesic torsion vanishes only along the lines of curvature. He then proves (i) that if two curves cut at right angles on a surface their geodesic torsions at their point of intersection are equal and opposite, (ii) that if two surfaces cut at right angles the geodesic torsion of their line of intersection is the same with respect to either surface. Dupin's theorem then follows in a few lines from the equations τ₁ + τ₂=0, τ₁ + τ₂=0, τ₂ + τ = 0. I was inclined to withdraw my proof after finding M. Fouchée's memoir, but I think its retention is justified by my use of the framework of elements, which makes obvious the torsion theorems proved at some length, though charmingly, by M. Fouché. In fact, all proofs of Dupin's theorem, with the exception of Herr Sommerfeld's, the substance of which I am giving in a later note, entail the establishing of equations of the type τ + τ₁ = 0, whether the geodesic torsions be actually designated as such, or occur analytically like the [qr. p], etc., of Cayley's proof, or as coefficients like the b, b′, b″ of Lord Kelvin's.

* I have read recently a paper by Herr Sommerfeld entitled “Geometrischer Beweis des Dupin'schen Theorems und seiner Umkehrung ” (Jahresbericht der deutschen Matematiker Vereinigung Band VI., Heft 1, 1897). Taking u, v, w to be the direction cosines of a Curvencongruenz, Herr Sommerfeld proves the relation by applying Stokes' theorem. The converse he establishes in a most simple and beautiful manner by the same means, and with the help of this Hülfssatz obtains Dupin's theorem. The proof of Dupin's theorem is so out of the common and can be put so briefly that I think I may give an account of it here. Let u, v, w be the components of the normal displacement which will transfer points on any surface (say AOB, Fig. 3) of one of the families to the consecutive surface. Then since OA, OB, OC remain at right angles after strain they must be the principal axes of strain at O. The strain can be analysed into three parts, a translation, a displacement normal to the strain quadric, and rotations , etc., about the axes of co-ordinates. The first two parts do not alter the directions of the principal axes, and since v, v, w are proportional to the direction cosines of a curve congruence orthogonal to a family of surfaces, so that the component rotation about OC is zero. Hence OC is an element of a line of curvature of each surfaoe on whioh it lies.

* It has been shown that if BD and CD′ do meet the condition is satisfied, and it is quite easy to show that the converse holds to the degree of accuracy stated.

* Journal de l'École Polytechnique, t. xvGoogle Scholar

† Liouville, t. xi.

‡ Journal de l'École Polytechnique, t. xxvGoogle Scholar

§ Comptes Rendus, t. lxxv.Google Scholar

* Quarterly Journal, vol. 22Google Scholar

* It may also be written

* Cayley's use of δ for is a little misleading, as the symbol is usually associated with an increment rather than a rate of change along the normal. Salmon, however, retained the notation, and an alteration would create confusion.

* The notation is sufficiently obvious: the symbol = is to be read “are proportional to.”

* It is easily found by actual expansion that

are given by Cayley and quoted by Salmon.

* It is hardly necessary to prove that are the curvatures of the principal normal sections considered positive when the corresponding centres of curvature are on the side of the tangent plane remote from the region into whioh the normal is drawn: R is, of course, always positive.