CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
451 
and on substituting the values of the sectional stresses from (43) and (44), in the first 
three of (11), we ought to reproduce the equations of motion in terms of the displace¬ 
ments which we have obtained from the variational equation. 
From (30), (3G), (40), and (42) it follows that these equations are 
p\2(u-X) + W^ 
(IK 
211 (ho 
da dz 
, , 1 dm 
^ dz ■*" 2a dcp 
- ® & 
+ S Ja Tfe + » 
\ dz 
dz 
(45). 
2 {y 
^ 6a d<p 
4vt ( 7 , “h "2 
ya dip 
da 
dm 
dz 
V — 
+ 
dvj 
d<p 
dydd I 
3a2 
cl'p , , dm 
V “h 2 T 
dz dz 
+ j ,1 M + I A') + y + M y _ 2 V, + Y 
\(t d<p ^ dz j oa d(p oft" \d(p 
(46). 
p\2{w — Z) — ^ E (X -h fx) 
I^ekI 
Oft J 
4?i23 , 4:nld r „ fZ-15 
-+ err ^ -f EdF + a y-^ 
ft oft" (_ dz-' ctep-' dz (((p 
2'nJd 4ftA" d fcl^ 1 Jct\ ^ d fdto n X 
3ft 3ft dz\clz 2a d(p] « ft 
2 pld d fdvj 2 p/r „ , •• „ 
Oft" a 
(47). 
If we compare these ec|uations with the equations obtained by substituting 
the values of the sectional stresses in the first line of ( 11 ), it will be found that 
they agree in every respect. 
10 . It will hereafter be necessary to consider certain problems in which the middle 
surface is supposed to experience no extension or contraction throughout the motion ; 
and it will, therefore, be necessary to obtain the recpiisite equations when this is 
supposed to be the case. . 
The conditions of inextensibility are 
cTj =: 0 , o'g = 0 , <7T = 0 ; 
or 
da 
dz 
= 0 , 
dv 
d<p 
+ 10= 0 , 
dft . dv 
. (48), 
3 M 2 
