90 TRANSFORMATION OF SURFACES BY BENDING. 



tying the problem. He has shewn that on every surface it is possible to find 

 two systems of curves intersecting at right angles, such that the length and 

 breadth of every element into which the surface is thus divided shall be equal, 

 und that an infinite number of such systems may be found. By means of these 

 curves he has found a much simpler expression for the specific curvature than 

 that given by M. Gauss. 



He has also given, in a note to his edition of Monge, a method of testing 

 two given surfaces in order to determine whether they are applicable to one 

 another. He first draws on both surfaces lines of equal specific curvature, and 

 determines the distance between two corresponding consecutive lines of curvature 

 in both surfaces. 



If by assuming the origin properly these distances can be made equal for 

 every part of the surface, the two surfaces can be applied to each other. He 

 has developed the theorem analytically, of which this is only the geometrical 

 interpretation. 



When the lines of equal specific curvature are equidistant throughout their 

 whole length, as in the case of surfaces of revolution, the surfaces may be 

 applied to one another in an infinite variety of ways. 



When the specific curvature at every point of the surface is positive and 

 equal to a', the surface may be applied to a sphere of radius a, and when the 

 specific curvature is negative = a" it may be applied to the surface of revo- 

 lution which cuts at right angles all the spheres of radius a, and whose centres 

 are in a straight line. 



M. Bertrand has given in the XHIth Vol. of Liouville's Journal a very 

 simple and elegant proof of the theorem of M. Gauss about the product of 

 the radii of curvature. 



He supposes one extremity of an inextensible thread to be fixed at a point 

 in a surface, and a closed curve to be described on the surface by the other 

 extremity, the thread being stretched all the while. It is evident that the 

 length of such a curve cannot be altered by bending the surface. He then 

 calculates the length of this curve, considering the length of the thread small, 

 and finds that it depends on the product of the principal radii of curvature 

 of the surface at the fixed point. His memoir is followed by a note of 

 M. Diguet, who deduces the same result from a consideration of the area of 

 the same curve ; and by an independent memoir of M. Puiseux, who seems to 

 </ive the same proof at somewhat greater length. 



