Page:Proceedings of the Royal Society of London Vol 4.djvu/137

Rh suit, to which they lead, equally indeterminate ; and that, therefore, when such result assumes the form ^, its true character is to be

tested by the equations that have led to it, after these have been modified by the hypothesis from which that form has arisen.

In a " Memoire surla Courbure des Surfaces/' (Journal de 1 Ecole Polytechnique, Tom. XIII.), Poisson has arrived at the conclusion, that the number of lines of curvature passing through an umbilical point is infinite, and that those selected by Dupin differ from the others only by satisfying an additional differential equation ; those others equally satisfying the conditions of a hue of curvature. These are precisely the conclusions arrived at by the author. As, however, he considers that the mode of investigation pursued by Poisson is pecuHar and ill adapted to the objects apparently in view, namely, to reconcile the results of ]\Ionge and Dupin and to remove their obscurities, he was induced to investigate some of the more important properties of curve surfaces, by a method somewhat dif- ferent from that usually emploved.

Adopting Z ='F (X, Y)

as the general equation of any surface ; by attributing to X, Y, Z, increments x, y, z, and assuming that the axis Z coincides with the normal to the surface, or that the plane x y parallel to the tangent plane, an equation equivalent to, and nearly identical with, Dupin' a equation of his indicatrix, is readily deduced. From this are imme- diately derived some properties of the radii of curvature, first shown by Dupin ; and likewise the theorem of jMeusnier. The author then enters upon the subject of the hues of curvature.

From the equations

A = 0, B = 0,

of the normal to the surface at a point on it, the equations of the normal at a point near to the former are determined. That these normals may intersect, which is the condition giving the directions of the lines of curvature, the two sets of equations must simulta- neously exist ; and hence are deduced the differential equations of condition for the lines of curvature,

d X dy d X ' d x d y d x

By this method, which fundamentally is not very different from that of Monge, substituting the usual expressions for A and B, the equation that determines the directions of the Hnes of curvature is deduced, in the form in which it had been previously given by Monge and Dupin.

This final equation becoming at an umbihc of the form, idv\i I dy\

in which ^ may be indeterminate, the author inquires how this ia» d £ ^