The Rev. J. H. Jellett on the Properties of Inextensille Surfaces. 371 



denoting by (cn)„ the normal displacement of the point on the surface. Equa- 

 tion (0') becomes, therefore, 



In = (Sn)o + i ^Cdn. 



Now the definite integral 



is evidently a small quantity of the second order ; if therefore we neglect 

 quantities of the third order, we shall have 



In = (Bw)o. 

 Hence we infer that — 



In all possible displacements of a thin membi'ane or lamina which is very 

 slightly extensible, the normal disjjlacements of p)oints situated on the same normal 

 to the surface are equal. 



This would also follow from the nest theorem. 



Substituting in the first two equations (N') for a, ^,7 their values in terms 

 of j9 and q, we have 



dlx' diz' 1 



fdZz' dZxf dly'\ .. 

 f)['d^-P-d^-^'W) = '^' 



dn ^ dn /(I +/»^+ ?% xuct u.t uu, / ,p,s 



diif dlz' 1 f diz' _ d^_ di^\ _ .J. 



dn ^P dn ^ ^/(l +p' + f) \ dy ^ dy ^ dy ) ~ 



Now it is plain that without altering the form of these equations we may 

 substitute, in the last three terms of each, ex, ly, Iz for Ix', cy\ Iz' . For this 

 substitution merely amounts to the addition of quantities of the same order as 

 lA, iB, to the right-hand members of these equations. Again, referring to 

 p. 348, we have 



ddz dEx dhj 

 d^-P-d^-'^'d^ = 'P^ 



dZz dlx dly _ 

 dy dy " dy ~ 



Making these substitutions in equations (P'), we have 

 VOL. xxn. 3 c 



