[ 491 ] 



XVI. Tlie Smnll Free Vibrations and Deformation of a Tlnn Elastic Shell. 



By A. E. H. LOVE, B.A., Fellow of St. John's College, Cambridge. 



Communicated by Professor G. H. DARWIN, F.R.S. 



Received January 19. Road February 9, 1888. 



CONTENTS. 



PAOI. 



1. Historical introduction. Poissox ; KIKCHHOFF'S first theory of plates; KiRCHBorr's 



second theory ; BOUSSISESQ ; DE ST. VENANT 491 



2. Theory of the present paper for thin shells ' . 496 



3. Internal strain in an element of the shell 499 



4. Geometrical theory of small deformation of extensible surfaces 605 



5. Equations of motion and boundary-conditions 512 



6. Possibility of certain modes of vibration .VJ' 



7. Vibrations of spherical shell 527 



8. Vibrations of cylindrical shell 538 



9. Summary of results 543 



1. Historical Introduction. 



I PROPOSE, in the first place, to give a brief account of the principal theories of the 

 vibrations and flexure of a thin elastic plate hitherto put forward, and afterwards to 

 apply the method of one of them to the case when the plate in its natural state has 

 finite curvature. 



Passing over the early attempts of Mdlle. SOPHIE GERMAIN, the first mathematician 

 who succeeded in obtaining a theory of the flexure of a thin plane plate was POISSON. 

 In his memoir* he obtains the differential equation for the deflection of the plate, 

 which is generally admitted, and certain boundary-conditions, which have met with 

 less general acceptance. The idea of POISSON'S method may be simply stated. The 

 plate being very thin, we may expand all the functions which occur in the equations 

 of equilibrium and boundary -conditions in powers of the variable expressing the 

 distance of a particle from the middle-surface in the natural state, then, taking only 

 the terms up to the third order, we obtain the differential equations for the determi- 

 nation of the displacements which are generally admitted. The meaning of POISSON'S 

 boundary-conditions is as follows t; Suppose the plate to form part of an infinite 



" Mi'inoiro sur 1'Eqnilibre et le Mouvement des Corps elnstiqnes," ' Paris A cad. Mem.,' 1829. 

 t Cf. THOMSON and T.ur, Natural Philosophy,' part 2, pp. 188-9 



3 R 2 2f,.11.8 



