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XLIII. On the Application of Force ivithin a Limited Space, 

 required to produce Spherical Solitary Waves, or Trains of 

 Periodic Waves, of both Species, Equivoluminal * and 

 Irrotational, in an Elastic Solid. By Lord Kelvin, 



g.c.v.o., p:r.s.e.\ 



§ 1. npHE complete mathematical theory of the propagation 

 JL of motion through an infinite elastic solid, including 

 the analysis of the motion into two species, equivoluminal and 

 irrotational, was first given by Stokes in his splendid paper 

 " On the Dynamical Theory of Diffraction " §. The object 

 of the present communication is to investigate fully the forcive 

 which must be applied to the boundary, S, of a hollow of any 

 shape in the solid, in order to originate and to maintain any 

 known motion of the surrounding solid ; and to solve the 

 inverse problem of finding the motion when the forcive on, 

 or tha motion of, S is given, for the particular case in which S 

 is a spherical surface kept rigid. 



§ 2. Let f, 77, f denote the infinitesimal displacement at any 

 point of the solid, of which (x, y, z) is the equilibrium 

 position. The well-known equations of motion || are 



d 2 % n , . v dS 

 P^2 ={k + ±n) Tz + n\/% 



where 8 denotes ~r +~r + i~- Using the notation of 

 ax ay dz ° 



Thomson and TaitU for strain-components (elongations ; and 



distortions), e,f,g; a, b, c; we have 



e ~dx? J ~dy' y dz 



a~ 



dz dy ' dx dz 5 cly dx 



(2); 



* By " equivoluminal '' I mean every part of the solid keeping its volume 

 unchanged during the motion. 



t Communicated by the Author : read before the Royal Society of 

 Edinburgh, May 1, 1899. 



§ Stokes, ' Mathematical Papers/ vol. ii. p. 243. 



|| See my paper " On the Reflexion and Refraction of Solitary Plane 

 Waves, &c," Proc. R. S E. Dec. 1898, and Phil. Mag. Feb. 1899. 



% Thomson and Tait's ' Natural Philosophy,' § 669, or ' Elements,' 

 §640. 



