88 TRANM>I;MATION OF SURFACES BY LKSLINO. 



to be satisfied during the whole process we must have another, which is the 

 condition for " Permanent Lines of Bending." 



The use of these lines of bending in simplifying the theory of surfaces is 

 the only part of the present method which is new, although the investigations 

 connected with them naturally led to the employment of other methods which 

 had been used by those who have already treated of this subject. A state- 

 ment of the principal methods and results of these mathematicians will save 

 repetition, and will indicate the different points of view under which the 

 subject may present itself. 



The first and most complete memoir on the subject is that of M. Gauss, 

 already referred to. 



The method which he employs consists in referring every point of the 

 .surface to a corresponding 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 curvature " 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 



