TRANSFORMATION OF SURFACES BY BENDING. 89 



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, x 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 cur- 

 vature may be found in terms of p and q when the surface is known. 



In this way it is shewn 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 shewn 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 the two systems of lines. One case is when the lines of 

 the first system are geodesic, or " shortest " lines having their origin in a point, 

 and the second system is drawn so as to cut off equal lengths from the curves 

 of the first system. 



The angle which the tangent at the origin of a line of the first system 

 makes with a fixed line is taken as one of the co-ordinates, and the distance 

 of the point measured along that line as the other. 



It is shewn that the two systems intersect at right angles, and a simple 

 expression is found for the specific curvature at any point. 



M. Liouville (Journal, Tom. xu.) has adopted a different mode of simpli- 



VOL. I. 12 



