The Rev. J. H. Jellett on the Properties of Inextensible Surfaces. 359 



We conclude, therefore, as before, that any one of these curves may be fixed 

 without destroying the flexibility of the surface. The reason for the name 

 " curve of flexure" is thus explained. In fact we see that these curves, when 

 fixed, allow the surfoce to bend round them, the flexure commencing at the 

 curve itself We shall presently show that this property is peculiar to the curves 

 of flexure as above defined. 



2. When the fixed curve is not a curve of flexure, the reasoning before 

 given in the case of developable surfaces will show that o,z', c^, B2, and all their 

 differential coefficients, vanish for the fixed curve. If, therefore, these func- 

 tions retained throughout the same form it is plain that the value of each could 

 be no other than zero. Before proceeding to consider how far this conclusion 

 is modified by a change in the forms of the functions, we shall prove the fol- 

 lowing theorems, which are essential to our purpose. 



(I.) If the functions which represent the displacements of an inextensible 

 surface have different forms at different points of the surface, the parts of the 

 surface for which these functions retain the same forms are bounded by curves 

 of flexure. 



This theorem is proved by reasoning nearly identical with that of p. 354. 

 For, if possible, let the forms of these functions change in passing across a 

 curve which is not a curve of flexure. Let 



u = U, V = V, w = TF, 

 be the values which hold at one side of the curve, and 



M = U\ V = F, w = W, 

 those which hold at the other. Then it will appear precisely as in p. 354 that 

 if we form a third system of values, 



u^U-U\ v=r-V\ w=W- TP, 



this system will satisfy the equations (C), and will, moreover, be such that for 

 every point of the curve in question we shall have 



M = 0, V = 0, w = 0. 

 Since, then, the bounding curve is not a curve of flexure, and since U, F, W, 

 U', P, TP, are functions of determinate form, it is plain that we must have 



generally 



U-U' = 0, F- F' = 0, W- W = 0. 



