The Eev. J. H. Jellett on the Properties of Inextensible Surfaces. 355 



The two systems of vahics are therefore identical. Hence we infer the 

 theorem — 



If a curve he traced upon an inextensible surface., whose principal curvatures 

 are finite, and of the same sign, and if any given determinate motion be assigned to 

 this cwwe, the motion of the entire surface is determinate and unique. 



Thus, for example, it is easily shown that if the limiting curve be made 



rigid, the entire surface will become rigid also. 



II. "We shall next consider the case of developable surfaces, or those in 



which 



rt - s= = 0. 



This case may be subdivided into two, which require to be considered sepa- 

 rately. These cases are — 



1. When the fixed curve is either a rectilinear section of the surface or 

 the arife de rebroussement. 



2. When the fixed curve is not either of these. 



1. Let the fixed curve be a rectilinear section. Then it is plain that this 

 curve must satisfy the equation 



r + 2s)H + tnr = 0, 



which expresses the fact that the radius of curvature of the normal section 

 passing through this line, i. e. in the case of a developable surface, of the line 

 itself, is infinite. 



Hence the equations (N) become identically true, without supposing that 



dxdy ' dxdy 



It is plain, therefore, that the reasoning by which it was shown that the several 

 differential coefficients of w, v vanish for the fixed curve, is no longer appli- 

 cable, and that the several conditions of the problem may be satisfied without 

 supposing u, V to vanish at every point of the smface. "\^'e infer, therefore, 

 that, 



In a developable surface composed of an inextensible membrane, any one of its 

 rectilinear sections may be fixed without destroying the fiexibility office membrane. 



VOL. XXII. 3 A 



