228 Mr. A. B. Basset on the Difficulties of Constructing 



Transactions ' ; but the surfaces of the shell are expressly 

 assumed to be free from external pressures and tangential 

 stresses, although the extension of the middle surface is taken 

 into account. As the analysis in the general case of three- 

 dimensional motion is somewhat long and complicated, it 

 will be desirable to reproduce in a concise form so much of 

 the original work as is necessary to obtain the value of the 

 flexural couple when the motion is supposed to be in two 

 dimensions. We shall neglect the extension of the middle 

 surface, but shall suppose that the shell is subjected to 

 external pressure. 



Let v', 10' denote the tangential and normal displacements 

 of any point of the substance of the shell ; let o\/, o- 3 ' be the 

 extensions in these directions, and tx 1 the shearing stress. 

 Also let the unaccented letters denote the values of the 

 quantities at the middle surface, where r = a ; and let the 

 values at the middle surface of the differential coefficients of 

 the various quantities with respect to r be distinguished by 

 brackets. 



Let r = a + h r , and let 2A be the thickness of the shell. 

 Then, employing Thomson and Tait's notation for stresses and 

 elastic constants, we have 



, 1 /dv' , \ 

 f die' 

 , dv' v' , 1 dw 



w = —. 1 . 



dr r r d(f> \ 



> (7) 



Now 

 B' = (m + 'O ^ + ( m ~~ n ) a, 2 



= (m + 



-)-3+{(- + n)(J 3 ) + (m-,)(^)}A' + ..., (8) 



for since the middle surface is supposed to be inextensible, 

 <r 3 =0. 



Since B/ is some function of h and A', it follows from 

 Taylor's theorem that 



B'=A + A 1 A / + A 2 A' 2 + (9) 



where A,A X . . . are unknown functions of the displacements and 

 the thickness. Accordingly, equating coefficients and putting 



E=(m-7i)/(m + tt), (10) 



