Eliminating U, V, W, in turn between (5) and (6) we get U. = k (VW, - V,W), W, = k (UV, - 0,V), U,dx + Vndy + W..dx is a perfect differential function, it follows that (VW, - V,W) dx + (WU, - W,U) dy + (UV, - U,V) dõ, may be integrated by means of a factor; and this is all the information which the equations (5) and (6) give with respect to the intersection of the surfaces (1) and (2). By the ordinary condition of integrability by a factor, we have But since Udx + V dy + Wdz is a complete differential function we have du dV dŲ dW dVdWV - dy da da da da dy and similarly for U, V, W,; therefore the equation may be put under the form Again, as (4) is identically true we may differentiate it with respect to x, y, x separately, when we have dU, dv, dw, du d V dW + W x dx and similarly for the others. Hence equation (7) becomes The curve which is the intersection of any surface of (1) with any surface of (2) satisfies the equations U dx + V dy + W dx = 0, (10) By combining these we have VW-V,W=dx, WU, - UW,=dy, UV, -U, V = \dx; 1 being an unknown quantity: and by combining (10) with (4) we have also U, = u(Vdz-Wdy), V= (W20-Udx), W, = (Udy-Vdx), le being an unknown quantity. Hence equation (8) becomes U,dU + Vid V + W,d W = 0, or (Vdx-Wdy) dU + (Wdx – U dr) d V + (Udy - Vdx) d W = 0. But this is the equation to the lines of curvature on the surface (1), and from the symmetry of the equations (4), (5), (6) it is clear that a similar result may be obtained for the surfaces (2) and (3). Hence the three systems of surfaces intersect each other in their lines of curvature*. • This demonstration of Dupin's Theorem was communicated to me by Mr Leslie Ellis. Another demonstration of Dupin's Theorem, of great simplicity, has been given by Mr Thomson in the Cambridge Mathematical Journal for February , In Liouville's Journal de Mathématiques, Vol. v. p. 313, the reader will find a memoir on Curvilinear Co-ordinates by Lamé, in which are demonstrated many very curious Theorems respecting the curvature of orthotomic surfaces. Sect. 3. Singular points and lines in Surfaces. (1) be the equation to a surface; then the direction cosines of the tangent plane at a point x, y, z are dF F? dF dF ? If now we can find values of x, y, z which, satisfying equation (1), also make at the same time dF dF dF = 0, = = 0, = 0, the position of the tangent plane at the point in question will become indeterminate, since the direction-cosines then take the form: At such a singular point we shall then have generally not a single tangent plane but many, even an infinite number, in which case their ultimate intersections will form a tangent cone, the vertex of which will be the singular point in question. If the three equations (2) are satisfied by assigning certain relations between the variables, then the curve formed by the intersection of the surface (1) with that indicated by the relation between the variables which satisfies equations (2) is a locus of singular points, that is to say, it is a line in which two or more sheets of the surface intersect. If for possible values of two of the variables on one side of the singular point we find impossible values of the third variable, that point is a cusp. If the same occur at every point of the singular line, it is called an edge of regression (arète de rebroussement). Such for examples are the curves which are the loci of the ultimate intersections of the generating lines of developable surfaces. To determine the equation to the tangent cone (if there be one) at a singular point, or the angle made by the tangent planes at the same point of a singular line, we proceed as follows. Giving the same designations as before to U, V, W, U, v, w, u', u', u', we have, by differentiating equation (1), Ud& + Vdy + Wdx = 0. + 2u'dy dx + 20'dx dx + 2w'dx dy = 0. Now at a singular point, U = 0, V = 0, W = 0, and this equation is reduced to (u) d x* + (v) dy + (w) dx + 2 (u') dydz +2 (o') dx dx + 2 (w') dx dy = 0, (3) the bracketed letters indicating the values they take when we substitute for x, y, x their values at the point in question. If all the quantities u, v, w, u', u', w' vanish, we must proceed to another differentiation, but in the examples which we shall adduce this will not be necessary. Now the equation (3) gives a relation subsisting between the increments dx, dy, dx in the surface at the singular point. These are the same for the surface and for a straight line touching it at the point; and therefore equation (3) gives a relation between the increments dir, dy, dx on the tangent lines at the singular points, or since this relation is the same for all points of these lines, we may substitute x, y, s for dæ, dy, dx in (3), and we find (u) x + (v) y* + (W)x+2 (u')yx+2(0')*x + 2(w')xy = 0, (4) as the equation to the locus of the tangent lines at the singular point which is taken as the origin of co-ordinates. This equation, unless for particular values of the coefficients, is that to a cone of the second degree. If we had proceeded to the third differentials we should have found the equation to a cone of the third degree, and so on in succession. It may happen that the equation (1) may be decomposed into two factors of the first degree, and then it will represent two planes. The condition that this may be the case is (u)(v)(w) - (u)(u')?- (v) (0°)?– (w) (w')' + 2 (U') (v') (w') = 0. where p2 = x + y + z. |