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XII. On the Reflexion and Refraction of Solitary Plane 

 Waves at a Plane Interface behveen two Isotropic Elastic 

 Mediums — Fluid, Solid, or Ether . By Lord Kelvin, 

 G.C.V.O* 



§ 1. "rC'LASTIC SOLID" includes fluid and ether; except 

 J__i conceivable dynamics f of the mutual action 

 across the interface of the two mediums. Maxwell's electro- 

 magnetic equations for a homogeneous non-conductor of 

 electricity are identical with the equations of motion of an 

 incompressible elastic solid J, or with the equations expressing 

 the rotational components of the motion of an elastic solid 

 compressible or incompressible ; but not so their application 

 to a heterogeneous non-conductor or to the interface between 

 two homogeneous non-conductors §. 



§ 2. The equations of equilibrium of a homogeneous elastic 

 solid, under the influence of forces X, Y, Z, per unit volume, 

 acting at any point (x, y, z) of the substance are given in 

 Stokes' classical paper u On the Theories of the Internal 

 Friction of Fluids in Motion, and of the Equilibrium and 

 Motion of Elastic Solids/' p. 115, vol. i. of his ' Mathematical 

 Papers': also in Thomson and Tait's ' Natural Philosophy' 

 [§698(5) (6)]. Substituting according to D'Alembert's 

 principle, — pf, — py, —pi for X, Y, Z, and using as in 

 a paper of mine |] of date Nov. 28, 1846, V 2 to denote 



d 2 d l d 2 

 the Laplacian operator -7-3 + j— 2 + 'J^-ii we ^ n< ^ as the equa- 

 tions of motion x & z 



pS =( - k+ ° n)d £ +nv ^ 



* Communicated by the Author j having- been read before the Royal 

 Society of Edinburgh on December 19, 1898. 



t See Math, and Phys. Papers, vol. iii. art. xcrx. (first published 

 May 1890), §§ 14-20, 21-28; and particularly §§ 44-47. Also Art. c. 

 of same volume ; from Comptes Rendus for Sept. 16, 1889, and Proc 

 Roy. Soc. Edinb., March 1890. 



\ See ' Electricity and Magnetism,' last four lines of § 616, last four 

 lines of § 783, and equations (9) of § 784. 



§ Ibid. §611, equations (1*). In these put 0=0, and take in con- 

 nexion with them equations (2) and (4) of § 616. Consider K and u as 

 different functions of x, y, z ; consider particularly uniform values for 

 each of these quantities on one side of an interface, 'and different uniform 

 values on the other side of an interface between two different non- 

 conductors, each homogeneous. 



|| Camb. and Dublin Math. Journal, vol. ii. (1847). Republished as 

 Art. xxvn., vol. i. of Math, and Phys. Papers. 



