350 The Rev. J. H. Jellett on the Properties of Inea:tms'Me Surfaces. 



= ^175; = 0. (K) 



This equation is the analytical statement of Gauss's celebrated theorem, 

 namely, that 



In all the possible movements of an inextensible surface, the product of the 

 principal radii of curvature at every point of the surface is constant. 



Let /S be a portion of the surface bounded by any closed curve. Conceive 

 this curve to be referred to the surface of a sphere, by radii drawn parallel to 

 the normals, and let S' be the included portion of the spherical surface. Then, 

 if the radius of the sphere be supposed to be unity, 



dS 

 and therefore, 



^'~]\rr" 



»^=|5^ = 0. 



Hence, In all possible motions of an inextensible surface, the area of the sphe- 

 rical cun-e corresponding to any closed curve described upon the surface (denomi- 

 nated by Gauss the " curvatura Integra") remains constant. 



This is the second theorem of Gauss. 



5. We shall nest proceed to consider the effect o{ fixing any curve upon 

 the surface. The determination of the displacement of the surface in this case 

 will obviously depend upon the following analytical problem : — " To find three 

 functions u, v, w, which shall satisfy the partial differential equations, 



du 



ay dx 



dv 



u;t = 0, 

 dy 



and shall, moreover, have the values 



M = 0, u = 0, w = 0, 

 for all points of a given curve or portion of a curve." 



