CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
441 
whence, if p = [d-GsJdr), the last four of (12) become 
G, = inld[^ + 
U _ 2 
Xlg — 
3 nldp 
(14). 
Since the couples are proportional to the cube of the thickness, it follows from the 
fourth and fifth of (ll), that the normal shearing stresses Ng are also proportional 
to the cube of the thickness, and therefore the terms of lowest order in the expres¬ 
sions for the shearing strains are quadratic functions of h and li, since such 
functions when integrated through a section of the shell, give rise to quantities 
proportional to the cube of the thickness. This is consistent with the fundamental 
hypothesis. 
The next thing is to calculate the values of the quantities X, fi, p)- 
From the first and fifth of (6) we obtain 
x = ('-^1 = 
dr 
dm„ 
d“VJ 
dz^ 
and, since the terms in X are all multiplied by we may put 
X = 
d^vj 
dz^ 
(15). 
Similarly from the second and fourth of (6) we obtain 
11= — 
Lastly, 
\d(f>- 
d^w , 
2+^ 
E 
(o-i -f 0 - 3 ) 
p = 
din 
dr 
d^u 
or 
a \dr d(f) 
2 d^w 
+ 
dv 
dr d'. 
1 dll 
cd dip 
1 dv 1 du 
^ a dz dip a d,z a? dip 
(16). 
(17). 
We have, therefore, completely determined the values of the couples in terms of 
known quantities. 
We shall also require the values of {ddujdr^), {ddajdr), [d^zsjdr'), the first two of 
which we have denoted by X', ju,'; and the last of which we shall denote by p'. The 
values of these quantities can, by a similar process, be shown to be 
X'=:E 
dz~ 
1 
2^ E d^K E 
^ “ V + a* 
( 18 ), 
'P I 
p = -h 
^ a cv 
7^3 . 2E 
+ 
a dzdip 
3 L 
MDCCCXC,—A. 
