442 
MR. A. B. BASSET ON THE EXTENSION AND FLEXURE OF 
Equation (17) and the last of (18), combined with the third and fourth of (12), 
determine the values of M^, Mg. 
It will be desirable to point out at this stage of the investigation, that we have 
obtained materials for the complete solution of any problem in which and u are 
zero, and none of the quantities are functions of 2 ;. The boundary conditions at a 
free edge will be discussed in § 11, and the reader who does not wish to be troubled 
with the long analytical process of finding the potential energy and working out the 
variational equation of motion, may pass at once to § 10, and the following sections 
where certain problems of a fairly simple kind are discussed. 
7. We must now find the potential energy due to strain. 
By the ordinary formula, the potential energy of a portion of the shell is 
w = 2 III 
+ n + rjs.p + — 4 (cr/o-g' + cr^Vg' + o'gV/)]] (l + li'/a) cZ^'c/S (19), 
where the integration with respect to z and </> extends over the middle surface of 
the portion considered. In evaluating this expression we may at once omit ct/, 
for since they are quadratic functions of h and h', they will give rise to terms which 
are proportional to which are to be neglected. 
Since 
it follows that 
A' = A + h' 
+ . . . 
^ (m + 
1 + 
I an 
= (m + n) { Aa-* + i ¥ A 
, 2/d 
+ 3,.^ 
^dA'' 
V dr, 
from which it is seen that W is expressible in a series of odd powers of h. 
From ( 5 ) we obtain 
A = (1 — E) (cr^ -j- cTg) + 
(f) = (' - E) (A + 
= -E)(X'+ ,.') + 
m + n 
m + 11 
m + 71 
and, therefore, the portion of W per unit of area of the middle surface, which depends 
upon A', is 
