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



And it is easily seen that the same conclusion will hold if the fixed curve be the 

 arete de rebroussement of the developable surface. 



2. Let the fixed curve be neither a right line nor the arete de rebroussement. 

 Then since this curve does not satisfy the equation 



r + 2sm + <»r = 0, 

 we must have, as in the first case, 



_^ = -^ = 0. 



dxdy ' dd'dy 



All the reasoning of that case is therefore strictly applicable, and it will appear, 

 as before, that all the differential coefficients of u must vanish for the limiting 

 curve. Hence, if u preserve the same form, it can have no value but zero. 

 Now let it be supposed that u may change its form ; then it is easily seen that 

 the zero value of u can only change in passing across a curve whose equation is 



)■ + '2sm. + tm- = 0. 

 Every part of the surface, therefore, which can be reached from the fixed 

 curve without crossing either the arete de rebroussement or a rectilinear section, 

 is necessarily fixed. The remainder of the surface is capable of motion. 

 Hence we have the following construction : 



Let AB be a fixed curve drawn on the given membrane. 

 Draw through the extreme points A, B, the rectilinear sections 

 of the developable surface, and produce them to touch the 

 arete de rebroussement. Then it is evident, from the foregoing 

 analysis, that all that part of the surface which lies between 

 the two lines, and on the same side of the arete de rebrousse- 

 ment with the fixed curve, will itself be fixed. Beyond these 

 lines the surface is flexible. 



To determine more accurately the nature of the motion of 

 which the surface is capable, we shall now proceed to integrate 

 the equation (D), wliich, for a developable surface, is in gene- 

 ral possible. 



Since rt — s" — 0, equation (D) becomes in the present case 



d-u d-u d'-u _ , „ s 



dy" dxdy dx- ~ ' 



The equations of the characteristic are therefore 



