CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
479 
23. When the middle surface is inextensible, it has been shown by Lord Rayleigh* 
that the displacements are given by the equations 
u = — tKs sin d taiP ^ 6 
v= sin P taiP 6^ 
w = 'tKs (.9 + cos 0) taiP ^ 6 
(49). 
where 9 = 2, 3, 4 . . . and is a complex function of the time. From these equations 
it can easily be shown by means of (48) that 
II ^ Xs {s’’ — s) e‘'P tan* \ 0 /I I' 
a a sin^ 6 \pn 
(50). 
The value of the potential energy is given by the second line of (16) ; also by the 
first two of (8) and by (48) 
and by the last of (8) 
whence 
X=- =--- 
^ Pi « 
(51). 
p=- 2t 2 
A.^ (.$•’ — s) tan* ^ 6 
a siiF 6 
4«7d 
W = 3 ^ 2 siiPg ~ ^ sin S(j) taiP ^ 6}'^] (.52), 
which agrees wdth Lord Rayleigh’s result. 
Let us now suppose that a bell which consists of a spherical shell, bounded by a 
small cLcle whose latitude is is vibrating in such a manner that its middle 
surface does not undergo any extension or contraction throughout the motion. One 
of the boundary conditions requires that the flexural couple Gj should vanish along 
the circle of latitude which constitutes the free edge of the bell. By (7) and (51) 
G^ = -f ^ nh^ {X + E (X + /r); 
From (50) we see that Gj cannot vanish for any value of 9 except 9 = 0, that is, 
at the pole, provided 9 > 2. It, therefore, follows that a spherical bell whose edge is 
free cannot vibrate in this manner if the middle surface is supposed to remain 
absolutely inextensible throughout the motion. If, however, extension or contraction 
* ‘ Loudon Math. Soc. Proc vol 18, p. 4 (1881). 
