CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
469 
sa = f {t. S. + M, + N, 8 «. s») + a (.A 
2rt/i®E.^ dSK rfSKl . 
+ Y,— -JW + o ■ c j.\cifin\dcl(p 
3a do 3a sin 6 dcf) } 
+ |{M.8« + T,8„ + N,S.<.+ fiQi 
Lastly, 
Gti / 1 dSw ^ \ Ho fdS'w ^ 
0 „'U “ 
2nJd'EM dSK nlv’ld.v^ fZSKI 
3a sin^ # 3a (16]^'^ ' 
8U = /J11 f (X hu' + Y 8r' + Z hiv) (1 + h'[af dh' 
dS 
= 2ph (1 + 
+ \ph?^ 
Id 
3a' 
XSa + YSt + ZSi(;)c/S 
- - Z (SX + 
a do a sin 0 d<p ' ' 
_ fyJlL --- SiiW ZE8KUS , 
1J L ft V <1^ ^ ft \siu 0 d<}> / 
(19). 
( 20 ). 
19. We shall, as in the case of a cylindrical shell, denote the four lines of W by W^, 
W„ Wg, W,. Whence 
SW;^ = 4tnh (1 At^/a^) J j ('^Scr^ + ^ ctSsi) dS 
= Anh ^1 + "I" 2 ^ ^ 0 d(f) -\- 1(23 Sr? + -| t::t Sii) a d6 
Anh[l + ~j[j 
aa' 
Yq sin 6») - 2^ cos 0 + 1 ^ j §w 
+ \ ^ ^) + i ^ cos ^jSu — (H + 23) sin 0 Siv 
ad0dcf) (21), 
from which we obtain the approxima.te ec[uations 
ArdiM, Tg = AnhM I 
Ml = Mg = 2w/inT J 
( 22 ), 
pu = 
pv = 
2n 
a sin 6 \ dO 
2n 
a sin 6 [ dcj) 
^^CasinO-J3cos» + i~| +pX 1 
^ ^ (ot sin ^ OT cos + pY > 
9.01 
pw = -(^ + 23) + pZ 
(23). 
