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

 dTox' cUz' ip 



dn '■ dn v'(l+/>- + 5'^) 



Integrating these equations between the limits and w, and neglecting as before 

 quantities of the third order, we have 



S {x'-x) +ph (z'-z) + ^(i^'J.^^.^ = 0, 

 But since 



(Q') 



these equations may evidently be written 



S\x'~x+p{2'-z)\ = 0, 



Hence recollecting that the points xyz, x'y'z\ were originally on the same nor- 

 mal, we have still, after the displacement, 



al — X + p (z' — z) —0, 

 y'-y + q(z'-z) = 0. 



We infer, therefore, that — 



In every possible displacement of a thin membrane or lamina whose extensibility 

 is very small, all points which icere originally situated on the same normal to the 

 surface will remain so after the displacement. 



This important theorem, which is assumed as an hypothesis by most writers 

 on the equilibrium of elastic lamina3, is thus established, independently of any 

 theory of molecular force, as a mathematical consequence of the small amount 

 of extensibility which is possessed by the lamina. 



It may be well, before concluding, to say a few words in explanation of the 

 rule which we have followed in the rejection of small quantities. 



Small quantities of the first order, as S.c, &c., have been retained throughout. 



