The Rev. J. H. Jellett on the Properties of Inextmsible Surfaces. 347 



The equation for w may readily be deduced from (C). Differentiating the 

 first of these equations with respect to y, and the second with respect to x, and 

 subtracting, we have 



d'V „ dw dw ds 



-J-, = 2s J r-j-+w-r-. 



da? ax dy dx 



Differentiating this equation with respect to y, 



d'v „ d-w d^w „ ds dw d^s 



= 26' -T—, r -T-^ + 2 -7- -J- + w - 



dx'^dy ~ dxdy dy- dy dx dxdy' 



Again, differentiating the third of equations (C) twice with regard to x, 



d^v d-w _ dt dw d'H 



-t -r-, +2-5--T- + M; 



dx^dy dx- dx dx dx- 



Subtracting these equations one from the other, we find, 



d^w d'w d-w _ 



dy^ dxdy dx- 



Some interesting results followed at once from the fundamental equations. 

 Thus, for example, if the displacements of the surface be all parallel to the 

 same plane, we shall have, taking this plane for the plane of xy, 



10 = 0. 



The equations (C) are thus reduced to 



du ^ du dv ^ f^f n 

 -T- = 0, -^ + ^ = 0, -r = 0- 

 dx dy dx dy 



Integrating this system of equations, we find, without difiiculty, 



u = A + By, V = C - Bx; 

 or since Iz = 



Ix = A+By, ly=C- Bx; 



A, B, C, being constants. These equations express the following theorem : 



If the displacements of an inextensible surface he all parallel to the same plane, 

 the surface moves as a rigid body. 



More generally, if we make 



w = Iz = ax — by + e, 

 VOL. xxn. 2 z 



