Royal Society. 2 1 3 



Polytechnique, Tom. XIII.), Poisson has arrived at the conclusion, 

 that the number of lines of curvature passing through an mubilical 

 point is infinite, and that those selected by Dupin differ from the 

 others only by satisfying an additional dilFerential equation ; those 

 others equally satisfying the conditions of a line 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 peculiar and ill adapted to the objects apparently in view, 

 namely, to reconcile the results of Monge 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 employed. 



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 jjlane x y is parallel to the tangent 

 plane, an equation equivalent to, and nearly identical with, Dupin's 

 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 Meusnier. The author then 

 enters upon the subject of the lines of curvature. 



From the equations 



A = 0, B - 0, 

 of the normal to the surface at a point on it, the equations of 

 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, 



dA dA dy dB dB dy 



dx dy ax dx dy dx 



By this method, which fundamentally is not very different from that 

 of Monge, substituting the usual expressions for A and B, the equa- 

 tion that determines the directions of the lines of curvature is de- 

 duced, in the form in which it had been previously given by Monge 

 and Dupin. 



This final equation becoming at an umliilic of the form, 



in whicli - may be indeterminate, the author inquires how this in- 

 determinate form will affect the equations of condition. As by this 

 supposition, these arc reduced to equations from which would result 

 the conditions that would render all the coefficients of the determi- 

 ning equation 0, it is inferred that — ^ vnist be indeterminate, and 



d X 



that therefore, at an umbilic there issue lines of curvature in all di- 

 rections. 



