The Eev. J. H. Jellett on the Properties of Inextensible Surfaces. 373 



Small quantities of the second order are rejected in the expressions for 

 daf, dy', dz', (H'), because the retention of these quantities would leave the 

 form of the equations (M') and (N') altogether unchanged. 



Small quantities of the third order are rejected in equations (0') and (Q'), 

 because the differences 



In — {cn)n, hx' — Ix, &c. 



ought properly to be of the second order. If we retain these terms we may 

 enunciate the foregoing theorem rigorously as follows : 



If a membrane which is hut slightly extensible receive a finite displacement, the 

 separation of a7iy pioint from the normal drawn through the corresponding point 07i 

 the surface, is indefinitely small, as compared with the distance of these points from 

 each other. 



With respect to the comparative magnitude of the two small quantities 

 i and n, depending respectively upon the extensibility and the thickness of the 

 lamina, it may have been observed that throughout the preceding discussion 

 they have been treated as quantities of the same order. Let us consider what 

 would be the effect of a violation of this rule. 



As the thickness of the membrane is not supposed to be insensible, we can- 

 not suppose n to be indefinitely small as compared with i, without assigning to 

 the membrane an amount of extensibility not indefinitely small. This would 

 remove it from the class of substances which we have been considering. 



If we had supposed i to be indefinitely small as compared with n, we should 

 not have been justified in rejecting ncla, nd^, ndy in forming the expressions 

 (H'j, p. 368. Our investigation would not, therefore, have differed in any ma- 

 terial respect from that of the displacements of a body of finite dimensions and 

 of an indefinitely small amount of extensibility ; and in such a case it would 

 readily appear from the discussion of p. 365 that the body would be, q. p., rigid. 

 We see then that — 



No membrane can be flexible ichich does not possess an amount of extensibility 

 finite, as compared ivith its thickness. 



It is, perhaps, superfluous to add, that it is not necessary to the truth of the 

 preceding theorems that the membrane should be absolutely or approximately 

 inextensible by any imaginable force. It is sufficient for our purpose if the 



2 c2 



