354 The Rev. J. H. Jellett 011 the Properties 0/ Inexfensible Surfaces. 



Eeplacing ii, v, w, by their values in terms of 2x, 8y, dz, we have for every point 

 of the surface 



5.r = 0, fy = 0, hz = 0. 



Hence we infer the following theorem : 



If any curve be traced upon an inextensible surface, whose principal curvatures 

 are finite and of the same sign, and if this curve be rendered immovable, the entire 

 surface will become immovable also. 



More generally, let it be required to determine a system of values of u, v, w, 

 which shall satisfy the equations (C), and which shall have at all points of a 

 given curve, or part of a curve, the given values 



U = Ui, V = Vi, W = Wi. 



Then it is easy to show from the foregoing discussion, that there is but one such 

 system. 



For, if possible, let there be two systems of values, 



u=U, v=V, w= W, 

 u^U, v=V', w= W, 



which satisfy the given conditions. Then since the equations (C), which these 

 two systems of values are supposed to satisfy, are linear, it is plain that if we 

 form a third system, 



u=U-U', v= V- V, w = W - W, 



this system will also satisfy equations (C). But as the values of u, v, w are 

 given for the limiting curve, the two assumed systems must be coincident 

 throughout tliis curve, and therefore we must have for all its points, 



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



Now we have seen in the foregoing discussion that if u, v,whe a. system of 

 values satisfying these two conditions, we must have generally 



u = 0, V = 0, w — 0. 



Hence it is plain that at every point of the surface 



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



