CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
467 
where the integration with respect to S extends over the middle surface of the portion 
considered. Since 
we obtain 
A' 
— A -j~ 
dr j 
\ [in + n) I A'^ (1 + h'jaY dh' 
also 
= (w + 7i) <1 /z (1 + A^ -f A 
4/d 
fdA\ 
dr 
in + n 3a^ + ^3 + Kj^nY + | ¥ (X [xf 
+ ^/d(o-i + 0 - 2 ) (X'+ p,')+ —(a-i+ o- 3 )(X + g)j . . . . 
2n [ o-'jcr'g (1 + h'/aY dli = Anli (1 + ^ Idja^) cr^cr 2 + f rJi^Xp 
j _ /( 
and 
+ 3 ^2 d“ 1 ) + 
8 w/d , . 
^ (Xo -2 + pO-j) 
( 12 ). 
( 13 ). 
2n[ (cr\ + cr'g) cr'g (1 + h'jaYdh' 
J —h 
= inh{l + — E (o-i 4 o- 2 )| [cr^ 4 cr^)— f n/dE(X 4 p)® 
* 1 (a77,7?^ _ 
-f n¥^ (cTj 4 cTg) (X 4 p-)-^ d“ p) (^1 + <^ 3 ).( 14)5 
lastly 
r ^ 4 -'?? h ^ 
|?ij (1 4 h'laYdh' = nh ( 14-3 ¥/cd) 4 ¥ n¥p^ 4 \n¥Top 4 (15). 
Substituting from (12), (13), (14) and (15) in (11), the value of W per unit of area 
of the middle surface is. 
W = ‘Inh (^1 4 {(tY 4 0 - 3 ^ 4 E (o-i 4 0 - 2 )® 4 i 
-\-^n¥ {¥ 4 p3 -p E (X 4 P)' 4 Ip^] 
■Y^n¥ (^X'4Bp 4 4^/) 
^(T .(iG)- 
3 o 2 
