442 



undergo in consequence of the displacement of the surface. 

 From these expressions Gauss's celebrated theorem as to the 

 product of the principal radii of curvature follows at once. 



The author has next proceeded to consider the effect of 



fixing any curve, or portion of a curve, upon the surface. In 



this investigation it is necessary to consider severally the three 



classes into which surfaces are divided with respect to their 



curvature, namely — 



1 . Oval surfaces, or those in which the principal curvatures 

 have the same sign. 



2. Developable surfaces, in which one of the principal 

 curvatures vanishes. 



3. Concavo-convex surfaces, in which the principal curva- 

 tures have opposite signs. 



The author has obtained the following remarkable results : 



1. Oval Surfaces. 



If a curve, or portion of a curve, traced upon an oval surface 

 composed of an inextensible membrane, be rendered immovable, 

 the entire surface becomes immovable also. 



2. Developable Surfaces. 



I. If an arc of a curve {which is neither the arete de re- 

 broussement, nor one of the rectilinear generatrices) traced upon 

 a developable surface, be rendered immovable, all that part of the 

 surface which lies between the rectilinear generatrices, drawn 

 thi % ough the extreme points of the fixed curve, and on the same side 

 of the arete de rebroussement with the fixed curve, will become 

 immovable also. Beyond these limits the surface will have the 

 power of motion. 



II. The arete de rebroussement, or a rectilinear generatrix, 

 may in general be fixed without rendering any finite part of the 

 surface immovable. 



III. The rectilinear generatrices of a developable inextensible 

 surface are rigid. 



3. Concavo-convex Surfaces. 



In surfaces of this class there exist (as is well known) at 



