1888.] Bending and Vibration of elastic Shells. 105 



December 13, 1888. 

 Professor G. G. STOKES, D.C.L., President, in the Chair. 



The Presents received were laid on the table, and thanks ordered 

 for them. 



The following Papers were read : 



I. "Spectrum Analysis of Cadmium." By A. GRUNWALD, 

 Professor of Mathematics in the Imp. Roy. German Poly- 

 technic University at Prague. Communicated by Professor 

 LIYEING, F.R.S. Received November 26, 1888. 



[Publication deferred.] 



II. " On the Bending and Vibration of thin elastic Shells, espe- 

 cially of Cylindrical Form." By LORD RAYLEIGH, M.A., 

 D.C.L., Sec. R.S. Received December 1, 1888. 



In a former publication* " On the Infinitesimal Bending of Sur- 

 faces of Revolution," I have applied the theory of bending to explain 

 the deformation and vibration of thin elastic shells, which are sym- 

 metrical about an axis, and have worked out in detail the case where 

 the shell is a portion of a sphere. The validity of this application 

 depends entirely upon the principle that when the shell is thin 

 enough and is vibrating in one of the graver possible modes, the 

 middle surface behaves as if it were inextensible. " When a thin 

 sheet of matter is subjected to stress, the force which it opposes to 

 extension is great in comparison with that which it opposes to bend- 

 ing. Under ordinary circumstances, the deformation takes place 

 approximately as if the sheet were inextensible as a whole, a condi- 

 tion which, in a remarkable degree, facilitates calculation, though 

 (it need scarcely be said) even beading implies an extension of all 

 but the central layers." If we fix our attention upon one of the 

 terms involving sines or cosines of multiples of the longitude, into 

 which, according to Fourier's theorem, the whole deformation may 

 be resolved, the condition of inextensibility is almost enough to 

 define the type. If there are two edges, e.g., parallel to circles of lati- 



* 'London Math. Soc. Proc.,' vol. 13, p. 4, November, 1881. 



