and Refraction of Solitary Plane Waves. 185 



reciprocal by a, we have 



a _ sin i _ sin /, __ shy' _ sin,/, 



where u and ?/, are the propagational velocities of the distor- 

 tional waves, and v, v n those of the condensational waves in 

 the two mediums. If now we take 



b = acoti= \Z(ii- 2 —a 2 ) ; b l = acoti l = s/(u~ 2 — a 2 ) ; 

 c=acotj= \Z(v~ 2 — a 2 ) ; e, = acotj t = \/(v t ~ 2 — « 2 ); (18), 



we have for the arguments of/ in the five waves 



t— ax + by; t—ax — by; t — ax + b t y; t — ax — cy; t — ax + Cjy (19), 



§ 10. Following Green * in calling the two sides of the 

 interface the upper and lower medium respectively (and so 

 shown in the diagram), we have for the components of the 

 displacement in the upper medium 



g=blf(t — ax + hy)—blf (t — ax — by) + aJ'f (t — ax — cy) | 



r) = alf(t — ax + by)+al'f(t — ax-by) +cJ'f[t — ax-cy) J ^' '' 



and in the lower medium 



^bLfit-ax+b^-raJJit-ax + cy) | 

 rj = al,f(t — ax + b,y) — c t JJ(t — ax+ c,y) ) 



where I, I', I,, J', J p denote five constant coefficients. The 

 notation J' and J, is adopted for convenience, to reserve the 

 coefficient J for the case in which the incident wave is con- 

 densational, and there is no incident distortional wave. 

 There would be no interest in treating simultaneously the 

 results of two incident waves, one distortional (I) and the 

 other condensational (J). 



§ 11. We may make various suppositions as to the inter- 

 facial conditions, in respect to displacements of the two 

 mediums and in respect to mutual forces between them. 

 Thus we might suppose free slipping between the two : that 

 is to say, zero tangential force on each medium; and along 

 with this we might suppose equal normal components of 

 motion and of force ; and whatever supposition we make as 

 to displacements, we may suppose the normal and tangential 

 forces on either at the interface to be those calculated from 

 the strains according to the ordinary elastic solid theory, or 

 to be those calculated from the rotations and condensations or 

 dilatations, according to the ideal dynamics of ether suggested 

 in the article referred to in the first footnote to § 1. We shall 



* Green's ' Math. Papers/ p. 253. 



