436 
MR. A. B. BASSET ON THE EXTENSION AND FLEXURE OF 
A question 'which is of fundamental importance in the theory now arises, as to the 
way in which the A’s depend upon li. 
If R' were of the order of the square of the thickness, it is evident that A and A^ 
could not contain any powers of li lower than the second and first respectively, whilst 
Ao could not contain any negative power of h. The A’s are entirely unknown 
quantities, and as there appears to be no possibility of determining them by an a 
priori method, it seems hopeless to attempt to construct any theory of thin shells 
without the aid of some assumption whicli will enable us to get rid of them. If, how¬ 
ever, ive assume, as has been practically done by ptrevious ivriters, that, when the 
surfaces of the shell are not subjected to any surface forces such as pressures or 
tangential stresses, R' and also S' and T', so far as they depend on h and h', are 
capable of being expressed in the form 
-h + • . . Un + . . . 
where is a homogeneous n-tic function of h and h', the problem can be completely 
solved without attempting to determine by any a priori method the values of any 
unknown quantities, and upon this fundarnental hypothesis the theory of the present 
paper ivill be based. 
There is some direct evidence of the truth of this hypothesis. In the case of a 
plane plate of infinite extent, it can be proved to be true by means of the general 
equations of motion of an elastic solid and for the purpose of testing the liypothesis 
in the case of a curved shell,'I have recently investigated to a second approximation, 
so as to obtain the term in Id, the period of the radial vibrations of an indefinitely 
long cylindrical shell, by means of the general equations, and also by means of the 
theory of thin shells, and both results agree.! But far the most conclusive evidence 
in favour of the truth of the hypothesis is furnished by the results to which it leads; 
and I have, therefore, conducted the following investigation in such a manner as to 
furnish a test of the correctness of the final results, and consequently of the funda¬ 
mental hypothesis by means of which they are deduced. 
Having obtained the equations of motion of a cylindrical and a spherical shell in 
terms of the sectional stresses, all these stresses are then calculated by a direct method, 
with the exception of the tensions T^ Tg, which cannot be calculated directly, 
since they involve the unknown quantities A and Ag. After that the potential energy 
and the other constitutents of the variational equation are calculated, and the variation 
worked otit by the usual methods. The final result, as is always the case in such 
investigations, consists of a line integral and a surface integral, the former of which 
determines the values of the sectional stresses in terms of the displacements, and the 
latter of which determines in the same fonn the three equations of motion. Now, if 
the work and the fundamental hypothesis upon which the theory is based are correct, 
* Lord Raylbigh, ‘ London Math. Soc. Proc.,’ vol. 20, p. 225 ; see also vol. 21, p. 33. 
t ‘London Maih. Soc. Proc.,’ vol. 21, p. 53. 
