The Eev. J. H. Jellett on the Properties of Inextensible Surfaces. 361 



and for the curve 





S = 0,ic. 



de ' do- ' de' 



For the point A, therefore, which is the intersection of these curves, both these 

 systems of equations must be satisfied. Putting, then, in equation (X), 



dw 



de 

 we have d-w 



= 0, 



do' "' 



7 = 0, 



dede 



and it is easily seen that neither P nor Q will become infinite ; and by follow- 

 ing the same reasoning with that of p. 352, we shall find that for the point A all 

 the differential coefficients of tc must vanish. Hence as the form of w remains 

 the same throughout the quadrilateral Abidci, we must have for the whole of 

 that quadrilateral 



v: = 0. 



Now it is evident that the reasoning which we have applied to the point A is in 

 every respect applicable to bi, and thus in succession to b^, is, &c. The value 

 of II- will therefore be zero for all points of a second curve of flexure Cidi. And 

 by pursuing the same method we see evidently that u- must vanish throughout 

 the whole of the quadrilateral ABDC. Hence, the direction of the axis of z 

 being indeterminate, we shall Jiave in general, 



8^ = 0, ly = 0, Ix = 0, 

 throughout ABDC. The whole of this quadrilateral is therefore fixed. We 

 shall now proceed to consider the general case. 



Let AB be any arc of a curve (not a curve of ? s .? 



flexure) traced upon the surface. Through A, B, draw 

 the curves of flexure, AC, AD, BC, BD. Then if AB 

 be fixed, the quadrilateral ACBD is fixed also. 



For whatever law or laws we suppose the dis- 

 placements to follow, it is plain that we may assume a 

 number of points, J,, 62. bj, &c. so close that one of 

 these displacements, w, for example, shall retain the 

 same form throughout each one of the quadrilaterals 



