CYLIN-DRIOAL AND SPHERICAL THIN ELASTIC SHELLS. 
Adding (24), (25), and (26), we finally obtain 
471 
SW2 = 
oa 
— {?E d- E (?G + d?)} ~ ip Sy + j — (ic sin ^) — df cos 0 
^ ^d(f) 
8w - l£ - ip 
d8i 
siu 6 d(f) 
— Sy 
a sin d cl(f) 
+ - 
4:nJi^ 
3a 
d- I n¥ 
-ipSu- {dP’d-E(i5d-d?’)}Syd- d-^cos^d-isin 0^) Sty 
E sin ^ — (is 
do ^ 
^ / 1 dBio « \ , 
- ^ (s.„ « ^ - i (w - S" 
+ d?)S» + E^(E + dF)5o 
a cW 
~ "i ^) d- (1 d- 2E) (is d- d?) sin 0 -fi {d^ cos 9) 
dd 
d- 2 (1 d- E) d? sin ^ d- cot ^ t 7 d- 
d(f) dd d(j) 
Sw 
d9d(f) (27). 
The expressions for Wg, may, as in the case of a cylindrical shell, be divided into 
two parts Wg', Wg", W/, W^". The values of SWg', SW/, may at once be written 
down from (21) by changing 13, trr into is', d?', 'P' and iE, dF, P respectively, and 
by altering the coefficient into ^nlP and d>n}Pl3a respectively. With regard to Wg" 
we have 
SWg" = f 7i]P [[(^ 8X'd- ^ Sp' + i Sp') dS. 
Substituting the value of X' from (9) and integrating once by parts, we shall obtain 
dSK 
SX' dS E sin 9 d<l> 
E ^ (a sin 6) + aa sin ^ {2 S\ + E (8X + Sf.)} 
Treating the other terms in a similar manner, we shall finally obtain 
\2nh^^^d3K 
d9 d(f>. 
SWg" = 
I 3a 
dO 
d- 
w/dECT dSK 
3a siu 6 d(j) 
+ 1 
r2ra/dEi3 Ti/dEro dgK 
I 3a sin 0 d(f> 3a d0 
a sin 9 d(f) 
a d9 
2n/d 
3fd 
E 
d 
sin 6 dO 
sin 9) — 33 cos 9 2 
dijs 1 dSK 
+ H 
E /d23 
dvr . 
AnlP 
:\ dgK 
d(p 
d(j) J d9 
d- Ea (^ d- 33) (SX d- Sp) 
dS 
A-7) 7? ^ r r 
--^jj (^SXd-33Spd-it^Sp)^/S. 
