CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS, 
475 
, A2\-- . hm dK 472 ch^ ^ 
2{l + Z^) ^ + T7-Y-. 7. - - 2 (1 + ^.) Y [ pa sm d 
oa sin 6 d(f) Zcr sin Q dcj) 
A 
dd 
r dd3 .id, • n\ , ^ n 
" [~d^ + i sm 0) TIT cose 
+ h74(AsinY>) + ^ 2^008 0 
-AEA(E+d?) + |rf{a'+i^| 
+ A {# + i ™ + + A 
( 3 G). 
72 
4/r 
2 ( i + 3 ^) - Z) — i (\ + p) - — EK j. pa sin ^ 
= — An (E + 23) sin ^ + 
3a 
47iA 2 r d^ 
'6a dd' 
+ A 
1 A-iF 
sin 6 d'cf)^ dd “*" ^ 
Q'vi ^2 
- I n¥ (IE' + d?') sin 0 - F- (IE + dF) sin 0 
- (IE sin ^) + (1 + 2 E) (IE + dF) sin 0 
dp d'^p 1 
oa 
2ph^ \ d , • r\\ , d^ . „ 
(37). 
The correctness of these equations may be tested by substituting the values of the 
sectional stresses from (33) and (34) in the first three of (5), when it will be found 
that we shall reproduce (35), (36), and (37). 
20. The boundary conditions for a spherical shell may be investigated in exactly 
the same manner as in the case of a cylindrical shell, by means of Stokes’ theorem ; 
for in the present case the theorem may be written 
f(fA) + i(f 1 ^) ® ’ 
the intonation extending round any curvilinear rectangle bounded by two meridians 
and two parallels of latitude. If, therefore, in the figure we apply to the side AD 
the stresses, 
M/ = HVa, = = 
" asm. d dcf) '• 
to the side BD the stresses 
M,' = H'/a, N/ = " , H,' = - H'; 
a aa . 
and to the sides OB, OA, corresponding and opposite stresses respectively, the 
preceding integral becomes 
3 p 2 
