TRANSFORMATION OF SURFACES BY BENDING. 451 



The method which he employs consists in referring every point of the surface to a corre- 

 sponding point of a sphere whose radius is unity. Normals are drawn at the several points of 

 the surface toward the same side of it, then lines drawn through the centre of the sphere in the 

 direction of each of these normals intersect the surface of the sphere in points corresponding to 

 those points of the original surface at which the normals were drawn. 



If any line be drawn on the surface, each of its points will have a corresponding point on 

 the sphere, so that there will be a corresponding line on the sphere. 



If the line on the surface return into itself, so as to enclose a finite area of the surface, the 

 corresponding curve on the sphere will enclose an area on the sphere, the extent of which will 

 depend on the form of the surface. 



This area on the sphere has been defined by M. Gauss as the measure of the " entire cur- 

 vature" of the area on the surface. This mathematical quantity is of great use in the theory 

 of surfaces, for it is the only quantity connected with curvature which is capable of being 

 expressed as the sum of all its parts. 



The sum of the entire curvatures of any number of areas is the entire curvature of their 

 sum, and the entire curvature of any area depends on the form of its boundary only, and is not 

 altered by any change in the form of the surface within the boundary line. 



The curvature of the surface may even be discontinuous, so that we may speak of the 

 entire curvature of a portion of a polyhedron, and calculate its amount. 



If the dimensions of the closed curve be diminished so that it may be treated as an element 

 of the surface, the ultimate ratio of the entire curvature to the area of the element on the 

 surface is taken as the measure of the " specific curvature" at that point of the surface. 



The terms " entire" and " specific" curvature when used in this paper are adopted from 

 M. Gauss, although the use of the sphere and the areas on its surface formed an essential part 

 of the original design. The use of these terms will save much explanation, and supersede 

 several very cumbrous expressions. 



M. Gauss then proceeds to find several analytical expressions for the measure of specific 

 curvature at any point of a surface, by the consideration of three points very near each other. 



The co-ordinates adopted are first rectangular, ,v and y, or x, y and z, being regarded as 

 independent variables. 



Then the points on the surface are referred to two systems of curves drawn on the 

 surface, and their position is defined by the values of two independent variables p and q, such 

 that by varying p while q remains constant, we obtain the different points of a line of the 

 first system, while p constant and q variable defines a line of the second system. 



By means of these variables, points on the surface may be referred to lines on the surface 

 itself instead of arbitrary co-ordinates, and the measure of curvature may be found in terms 

 of p and q when the surface is known. 



In this way it is shown that the specific curvature at any point is the reciprocal of the 

 product of the principal radii of curvature at that point, a result of great interest. 



From the condition of bending, that the length of any element of the curve must not be 

 altered, it is shown that the specific curvature at any point is not altered by bending. 



The rest of the memoir is occupied with the consideration of particular modes of describing 



