410.] If there are three systems of surfaces which mutually intersect each other at right-angles, every two of them trace on the third its lines of curvature. Let the three systems of surfaces be represented by the equations F1 (x, y, z) = fi (a), F2 (x, y, z) = f2 (a), F3 (x, y, z) = fs (a); (65) where a is a variable parameter by the variation of which each member of the several systems is formed; and where fi, f2, fs are symbols of functions and are such that the several systems intersect at right angles. Let U1, V1, W1, 1, v1, w1, ui, vi, w, and similar symbols, with the subscripts 2 and 3, represent the partial derived functions of (65) as heretofore; then, as the three surfaces intersect at right angles, U2 U3 + V2 V3 + W2 W3 = 0 U3 U1 + V3 V1 + W3 W1 = 0 U1 U2 + V1 V2 + W1W2 = 0 (66) Let dx, dy, dz be the projections on the axes of a length-element of the intersection of two members of F2 and F3; this is therefore perpendicular to a member of F1; so that and the x, y, z are subject to the relation given in the third equation of (66); so that differentiating we have U2 dv1 + U1d V2 + V2 dV1 + V1 dV2 + W2 dw1 + W1 d W2 = 0; (68) and replacing du1, ...... by their values, which are given in (61), Art. 360, we have U1 {U2 U3 + U3U2 + V2W3′ + V3 W2 + W2 V3′ + W3V2'} + V1 {U2W3' + U3W2′ + V2 V3 + V3V2 + W2U3′ + W3U2'} + W1 {UgV3' + U3V2′ + VqU3′ + V3Ug' + W2 W3 + W3 W2} = 0; (69) and this is in terms of second partial derived functions the condition that the line of intersection of two members of F2 and F3 should at the common point of intersection with the member of F1 be perpendicular to F. And two other similar expressions can be found, because the lines of intersection of members of F3 and F1, and of F1 and F2 respectively, are perpendicular to the members of F2 and of F3. If Ug UgU1+V2 V3 V1 + W2 W3 W1 so that the conditions requisite that the three systems of surfaces should intersect at right angles are U2 U3 U1+V2 V3 V1 + W2 W3 W1 +(V2W3+ V3 W2) U1 + (W2U3+ W3 U2) Ví′ + (U2V3 + U3V2)W1 = 0; (73) U3 U1U2 V3 V1 V2 + W3 W1 W2 +(V3W1 + V1 W3) U2 + (W3 U1 + W1 U3) Vg′ + (U3V1+ U1Vg)Wg'= 0; (74) U1 U2 U3 + V1 V2 V3 + W1 W2 W3 + (V1W2 + V2 W1) Ug′ + (W1 U2+ W2U1) Vg′ + (U1 V2 + U2 V1)w3′ = 0; (75) now from the first of these I propose to shew that an element of the line of intersection of two members of F1 and F2, at the point where they are intersected by the corresponding member of F3, is an element of a line of curvature of the member of F1. From (66) we have .. (V2 V3+W2 W3) U12 + (V1V2 + W1 W2) (V1 V3+W1W3) = 0; Let ds be the length-element on F1, which is the intersection of F and F2, and is perpendicular to F3; then also V2 dx + V2 dy + Wą dz = 0, U2 U1+V2V1+W2 W1 = 0; Therefore in (79) replacing U2, V2, W2, U3, V3, W3, by these proportionals, we have, after all reductions, and if we substitute in this expression the values of H, K, L given in (19), Art. 399, we have U1(K1-L1) dy dz + V1 (L1-H1) dz dr + W1 (H1-K1) dx dy = 0, (83) which is the same as (24), Art. 399, and is the differential equation to a line of curvature on F. It is also evident that we should have obtained the same equation if by means of (73) we had investigated the condition that the line of intersection of F1 and F3 should be perpendicular to F2 at the common point of intersection: and as (83) is a quadratic equation, in terms of the proportions of the length-element, its two roots refer to the two lines of intersection of F2 and F3 with F1; and therefore we conclude that the lines of intersection of F2 and F3 with F1 at the common point are the lines of curvature of F1. If processes in all respects similar are performed on (74) and (75), two equations with the subscripts 2 and 3, and similar in form to (83), will result: and therefore the members of F, and F1 will intersect all the members of F2, and the members of F and F will intersect all the members of F3, along their respective lines of curvature: and this is Dupin's Theorem. If therefore a system of surfaces of the form F1 is given, and if we can find any other two systems such as F2 and F3, the several members of which intersect all the other members of Fi and of each other at right-angles, then the lines of intersection are lines of curvature. 411.] Of this proposition we have one remarkable example. Let us call surfaces of the second order confocal, when their principal sections are confocal. Then three systems of confocal surfaces of the second order, which are severally an ellipsoid, a hyperboloid of one sheet, and a hyperboloid of two sheets, intersect at right angles; as we may thus prove. As the surfaces are confocal we may take their equations to be where >b>c2; μ2 >b2<c2; v2 <b2 <c2; so that the three equations represent severally an ellipsoid, a hyperboloid of one sheet, and a hyperboloid of two sheets. Now and therefore all the members of F2 and F3 intersect each other at right-angles. Similarly it may be shewn that all the members of F1 intersect at right-angles all the members of F2 and F3. Hence by Dupin's Theorem we infer that the lines of curvature on an ellipsoid are the lines of intersection of it with two confocal hyperboloids, which are respectively of one and two sheets; and thus the construction of the lines of curvature is reduced to the much more simple problem of the intersection of surfaces. 412] On the locus-surface of the centres of principal cur vature. Imagine at a given point on a surface the two lines of curvature to be drawn; and at every point along one of these lines of curvature imagine normal planes to be drawn touching it; these will by their intersection generate a developable surface, cutting the given surface at right angles; the normal line will be the characteristic of this developable surface; and the curve formed by their intersection will be its edge of regression, which will also be the locus curve of the centres of principal curvature whose section touches the line of curvature. Similarly, if normal planes are drawn touching the other line of curvature which passes through the given point, another developable surface will be generated which cuts the former at right angles; and there will also be another edge of regression, which is the locus-curve of these second centres of principal curvature. And as a similar process will hold true for all points of a surface, so will the series of developable surfaces arising from the first line of curvature cut at right angles each of the series arising from the second line of curvature, and thus will space be filled with developable surfaces intersecting each other at right angles; and as the edges of regression belonging to the first line of curvature continuously vary, so will they generate a surface-locus of all the corresponding centres of curvature. And similarly will another surface be generated by the other centres of curvature. We shall hereby obtain a surface of two sheets, to each of which the normal of the surface is a tangent; and any two planes drawn through the normal and touching the two sheets are at right angles to each other. Hence it appears that the surfaces which are the locus-surfaces of the two centres of principal curvature must have particular properties. They will be of two sheets; their algebraical equations therefore must be of even dimensions; these sheets must intersect at right angles. A surface therefore which does not fulfil these two conditions cannot be the surface of the centres of curvature; it may be the locus-surface of the centres of one principal curvature, but it will require another surface to be its conjugate, and this will be the locus-surface of the other centre of principal curvature. If at any point the two sheets of the locus-surface of the centres of principal curvature intersect each other, so that their edges of regression have a common point, the principal radii of curvature of the original surface corresponding to that point are equal, and there is an umbilic; and if there is a continuous locus of such points of intersection, the corresponding line on the original surface is a line of spherical curvature. See Art. 407. |