154 Mr. J. Walker on Cauchy's Theory of 



tional wave, propagated with velocity \/ , and for which 



the velocity-potential is <$> ; and a distortional wave, propa- 

 gated with the velocity \/-, and for which the components 

 V p 



of the displacement are u, v, w. 



Let the interface of the media be the plane of yz, and sup- 

 pose the first medium on the side of positive x. 



Considering only plane waves, which have the same period 

 of vibration 2tt/co and the same trace by + cz=0 on the inter- 

 face of the media, the values of </>, u, v, w satisfying the 

 equations may be taken as 



-p n 



<f> = n (a.x+bu + cz-L>f)V-i , ^ ^-a tl x+by+cz-tat) /-!*, 



9 s/^l + V-l 



_ a (ax + by-\rcz-(tit)"j^\ . a (~ax+by+cz-u)t)*/-l 



-p (ax+by+cz-ut)^— T , r» <-ax+by+cz—tat)'>/~ 



__. ri (ax+by+cz—afiJ^l , n (—ax+by+cz—taf)^-^l m 



where 



V m+n 7 V n ' 



and A, B, C ; A ; , B y , C, are connected by the relations 



Aa + B& + Cc=0, -A ; a + B ; 5 + C i c=0. 

 The corresponding values of £ , 97, f are accordingly 



t —J^ e (ax+by+cz-<ot)'S^Ji , A (~ax+by+cz-<at)>/^i , t» (ffl„#+ty+c*— atW^l " 



p _ {-a l fls+bg+cx-ut)*/'^\ 



« = B^ + ^ + ^" w ^" / ~ ri + B e (_aar+ ^ +C;jr_w ^ v/ ~ I H-B 5g( a ^+^+ c *- w/ )' v/ - r 



+ B C6 (- a '<*+^+ c «- w ^^-i 



Now within the medium the displacements are those due to 

 the distortional waves alone, and hence the values of f, 97, ? 

 are 



* The V — 1 is inserted for convenience. 



