CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
473 
have disposed of the terms in SW^, which involve h^, we shall write SW/ for the 
remaining portion which depends upon li, and the variational equation finally 
becomes 
§W/ + 8W/ -f- 8W3' + 8W/ + (1 + -5 h^jor) {n Su fi- v Sy “h w Bvj) dS 
I / cho 
cd \ cl6 
— 'Ill 
d Sw 
d0~ 
+72 
I / 1 dtv 
\sin 6 d<^ 
2v 
1 d Siv 
sill 6 d^ 
Sv 
d- e(|eK SKjdS 
Su 4- ^ Sy — (\ fi- u) Siy jc/S 
asm 0 dcf) ^ ' j 
a d0 
■iph^ ff n fdvj 
3 rt JJ 
a \ d0 
Sm + - (-} . qy — y) 8y + EK Siylc/S 
= 2phll + 
/P 
\ ' 3a^/jJ 
a \siii 0 d(j) 
XSu + YSv -i-ZSiv)dS 
2p¥ 
Ycd' 
/ 1 
\sin 
0 
-Sy +ZEaSK c/S 
+ j (Tj 8 a fi- N 2 Sw) a sin 6 dcp ^ (Tg 8 y fi- Sw) a dO 
(30). 
We have now got rid of all the terms involving the second differential coefficients 
of Su, Sv, Sw ; and all that remains to be done is to integrate by parts the terms which 
involve the first differential coefficients. Putting 
“ = + = + y=E(«EK-:,. + Z) (31), 
we have 
2pJ^Cf 
3a2 J J 
(X 
dSiv /3 dSvj 
"r ~ 
+ «y 8 KU^S 
d0 sin d d(j) 
~ olSw) a sin 6d((} -f- |(y 8 y + ^Sw) adO 
S‘fj [{% + rf! - -lysin d 8 «’ 
d(f) 
d(j) 
(-7S 
sill 0 
(32). 
Substituting the values of 8 W/, 8 W 3 ', 8 W 3 ', 8 W 4 ', and the right hand side of (32) 
in (30), and picking out the line integral terms, we obtain the following equations 
for the sectional stresses, viz., 
MDCCCXC.—A. 3 p 
