420 Sir William Thomson on the 



the condensational-rarefactional wave is irrotational, because 

 we see by (13) that an absolutely general expression for its 

 components, u—vf, v—v f y iv—w r , if denoted by u", v f/ , w", is 



u"=^, «"=^, w"=^ . . . (16), 

 dx ' df# ' dz 



where SP is any function such that 



V 2 ^=5 (17). 



Hence, as B satisfies (12), we have 



r5=AV 2 * (18); 



and we see, finally, that the most general solution of the equa- 

 tions of infinitesimal motion is given by 



u = u r + u", v=v' + v", iv = w f + w", . . (19): 



provided v! , v r , w' satisfy (14) and (15) ; and u" ', v ff , w" 

 satisfy (16) and (18). 



11. Let us now work out the general problem of reflexion 

 and refraction between two portions of homogeneous elastic 

 solid sliplessly attached to one another at a plane inter- 

 face, and having different densities, f, f 7 ; different rigidities, 

 B, B y ; and different values, A, A„ for the condensational- 

 rarefactional wave modulus. Thus, if a, /3 ; a n /5 7 denote 

 the velocities of the condensational-rarefactional, and of 

 the distortional, waves respectively in the two mediums, we 

 have 



/3=V(B/t), e,= </(B,IQf ' 



To avoid circumlocutions we shall suppose the interface hori- 

 zontal, and call the two mediums, or solids, the upper and the 

 lower respectively. Take OX vertically upwards ; and OY 

 horizontal to the right : and let the incident ray come from 

 the left obliquely downwards, in the plane YOX ; so that, if 

 % denote the angle of incidence, the equation of the wave-front 

 of the incident ray is 



x cos i+y sin %■= const. 



12. Consider first the case of vibrations perpendicular to 

 the plane of incidence. The medium being isotropic, no con- 

 densational waves can be generated at the interface. There 

 is therefore just one reflected and one refracted ray; all 

 vibrations are perpendicular to the plane of incidence ; and 

 all three waves are purely distortional. It is clear also that 



