CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
463 
notes which mainly depend upon the extension is usually high in comparison with 
the pitch of notes which mainly depend upon bending, and consequently the 
notes arising from the former cause, both on account of the smallness of their 
amplitudes and the highness of their pitch, would probably be so feeble in comparison 
with those which arise from the latter cause, as to be scarcely capable of producing 
any appreciable effect upon the ear. Judging from the usual course of such 
investigations, the probable form of the exact solution of the problems suggested 
at the end of § 13 would be that of an infinite series, the periods of the different 
components of which would satisfy a transcendental equation having an infinite 
number of roots ; but the preceding considerations point to the conclusion that the 
frequency of the gravest* note given by (60), viz., = ASmnh^j5pa^ (m -f- n), 
although perhaps not rigorously accurate, is a close approximation to the truth. 
Spherical Shells. 
15. The fundamental equations for a spherical shell can be investigated in precisely 
the same manner as in the case of a cylindrical shell. 
If u', v', iv' be the component displacements at any point of the substance of the 
shell in the directions, 6, r, the equations connecting the displacements and strains 
are 
cr , = 
a- 0 = 
(To = 
tJT 0 — 
3 — 
+ IV 
dv' 
1 
r \d6 
1 (Jl 
r \sin d d(f) 
dto' 
lir 
ff- ii cot 6 + lo' 
dw' dv' 
J7- + ^ 
r sin 6 d(j) 
du' u' 1 dw' 
dr r r dd 
1 /dv f .a , ^ 
~\Ta ~ cot ^ + -r-T — 
r \cW smO d(p 
> 
(1), 
* [The experiments of Lord Rayleigh, ‘Phil. Mag.,’ Jan., 1890, show that the effective pitch of a 
bell is usually not the same as that of its gravest tone; and, in the bells which he examined, the fifth 
tone in order was the one which agreed with the nominal pitch of the bell.—June, 1890.] 
