690 



Mr. E. V. Southwell on the 



and the equations (1) thus become, if we neglect infinitesi- 

 mals of the second order, 



dT 

 d<f> 



dW 



+ N' = 0; 



d<p \ d<f> 2 



ldoy 



a d<p 



+ N' = 0. 



1 



(IL-n^O; y 



J 



(5) 



From equations (5) a condition for stability can be derived, 

 provided that G' can be connected, accurately to terms in h 3 , 

 with the curvature of the distorted middle surface ; for by 

 eliminating T r and N' we obtain 



4( i4 50{? +(n '- n >H- • • (6) 



I shall now attempt to show that the requisite expression 

 for G' can be derived without the need for any assumption 

 as to the extension of the middle surface. We may proceed 

 by Mr. Basset's method, the strain quantities now denoting 

 the infinitesimal increments introduced by distortion. His 

 equation (8) becomes 



W = [m + n) cr 3 -f {in - n) a 2 + \ (m + n) 



dr ) 



so that if we write 



R'=A+A 1 // + A 2 / i /2 4-... 



(?) 

 (8) 



and 



m — n 



m-\-n 

 we have 



= E 



A 



<T C 



m 4- n 



E<j 2 , 



\ dr ) 



A> 



-E 



m + n \ dr J 



Mr. Basset's equation (12) then becomes 



fdaJ\ _ 1 f <fa A w 1 dh» I 



\ dr 1 a \ dcp m + n a a dd> 2 j 



m 



(10) 



