CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS, 
457 
also the condition of inextensibility gives 
dv 
d(j) 
+ IV 
0 . 
Eliminating Tg, and w, and substituting the value of from the first of (52), 
we obtain 
AmnJd / d 
Sped {jn + n) d(j) 
. (59), 
whence, putting 
we obtain 
^imnlds^ (s^ — 1)^ 
Sped {m + n) (s“ + 1) 
(60), 
which is the required result. 
If the cylinder is complete, s is any integer, unity excluded, but if tlie cross-section 
of the cylinder consists of a circular arc of length 2aa, s will not be an integer. Its 
values in terms of are the six roots of (60), but in order to obtain the frequency 
equation, the value of s in terms of the dimensions and elastic constants is required. 
The additional equations are obtained from the boundary conditions, which have to be 
satisfied along the straight edges of the shell, and these require that the tension Tg, 
the normal shearing stress N;^, and the flexural couple should vanish at the edges 
where ^ = d: “• 
Since 
^ SvmJd 
— ~ S (m + n) 
where 
_ 1 ! ePv dv\ 
the boundary conditions are obviously 
p = 0, 
elp, 
d(f) 
= 0 , 
4:7nnld d? 
+ 
edv 
Spa^iia + ii) defd dejidd 
= 0 . 
These conditions have to be satisfied at each of the edges of the shell where 
= rh and there are, therefore, six equations of condition; hence the six constants 
MDCCCXC.—A, 3 N 
