CYLINDRICAL ANT) SPHERICAL THIN ELASTIC SHELLS. 
455 
also hereafter appear, that observations of a precisely similar character apply to the 
corresponding equations which determine to a first approximation the extensional 
vibrations of a spherical shell. This portion of his paper therefore appears to be per¬ 
fectly satisfactory ; but that portion which involves the terms depending upon the 
cube of the thickness is open to criticism. 
In the first place, he appears to have employed a system of rectangular axes, con¬ 
sisting of the normal at a point on the middle surface, and the tangents to the two 
lines of curvature through that point. Now, although it is immaterial, so long as 
we confine our attention to infinitesimals of the first order, whether a quantity is 
measured along the tangents to three orthogonal curves or along the curves them¬ 
selves, yet when it is necessary to take into consideration infinitesimals of higher 
orders, which is always the case whenever an investigation involves changes of 
curvature, a method in which everything is referred to rectangular axes requires care; 
and on comparing the terms in A® in (24) with the corresponding terms in Mr. Love’s 
expression for the potential energy, it will be seen that he has omitted several terms 
which involve the extensions of the middle surface, which partly, although not 
entirely, arises from his having omitted the factor 1 + h'/a. It is not improbable 
that these terms may be small, but at the same time we are not at liberty to neglect 
them altogether ; for it is quite evident that a term such as S (2$/x) in the variational 
equation, will give rise to terms in the equations of motion and the equations giving 
the values of the sectional stresses, which do not involve the extension of the middle 
surface. 
In the second place, on comparing Mr. Love’s variational equation of motion* with 
my equations (25), (26), (28), and (29), it will be seen that he has omitted several 
terms in the expressions for 8C, §U, and 
In the third place he states (p. 521) that the extensional quantities “ cr^, o-g, ct- may 
not, in general, be regarded as of a higher order of small cjuantities than Xj, 
which are the quantities upon which the bending depends. The argument of Lord 
RAYLEiGHt appears to me to show, that at points whose distance from the edge is large 
in comparison with the thickness, the extensional terms are usually small in comparison 
with the terms upon which the bending depends. It must be obvious to every one, 
that a thin plate of metal or a steel spring can be bent with the greatest ease by means 
of the fingers, whereas the production of any extension of the middle surface which 
would be capable of measurement, would involve considerable muscular effort. These 
considerations indicate that when a thin shell is vibrating, the change of curvature is 
so greatly in excess of the extension of the middle surface, that notwithstanding the 
smallness of A® compared with A, the product A® (8p~^)^ is largelj; compared with 
* ‘Phil. Trans.,’ A., 1888, p. 514, equation (19). 
t ‘Roy. Soc. Proc.,’ vol. 45, p. 105. 
X The problem discussed in § 14 shows that the product may he of the order except in the 
neighbourhood of a free edge ; but in the equations of motion we have to deal with the quantities Tia and 
