466 
MR. A. B. BASSET ON THE EXTENSION AND FLEXURE OF 
The third and sixth of these equations satisfy the last of (5) as ought to be the case. 
Also, employing our previoiis notation, we see that 
Gj = - I nh? (d? + Ga = I (S + f) 
a ) 
Hi = — H3 = — I nh^ [p + 
( 7 ). 
Since the couples are proportional to the cube of the thickness, it follows from the 
fourth and fifth of (5) that the normal shearing stresses N^, Ng are also proportional 
to the cube of the thickness, and, therefore, that the shearing strains are 
quadratic functions of h and li. 
Employing our previous notation, the next thing is to calculate the quantities 
X, jx, p, p, p)'. We have 
in which equations we have omitted all quantities which vanish with h, because 
X, p, p occur in expressions which are multiplied by 1^. Similarly 
X' = 
p = 
2\ E , \ , E (PK 
- 7 - 
1 
2p E E/ 1 cPK , ! 
a « ' ' \siir 6 d(f)^ da / ■ 
2p 
a 
2E / , 
cP sin 6 \ d(f) dO dcf) 
J 
(9). 
18. The variational equation may be written 
SW + 8C == SU + 8E 
( 10 ). 
and we must now calculate the values of the four terms in it, and we shall begin 
with W. 
Since we may omit ct'j, and may, therefore, write 73 - for 73 - 3 , the potential energy 
of any portion of the shell is 
[ \{in-\-n) + w [ 73 '^ — 4 + cr'gcr',)}] (1 + h'jciY dh'd^ (11) 
J —h 
