The Rev. J. H. Jellett on the Equilihrium and Motion of an Elastic Solid. 191 



dydz = cos pdS', dzdx = cos qdS', dxdy = cos rdS ; 



we shall have 



\\ (P, cosjD + Pa COS 5- + P3 COS r) li,dS' 



+ J|(Qi cosp + Cti COS 5' + Q3 cosr) lt]dS' 



+ jl (Pi cosp + P2 cos q + Bi cos »■) ?f d5' 



+ l\ (X, 8§ + F. 8,; + Z, ?f ) e, rf5' = ; 



where A', , }', , Zi are forces acting solely at the surface of the medium. The 

 mode of treating this equation in the several cases which may occur having 

 been fully given by Mr. Haughton, I do not think it necessary to pursue this 

 part of the subject further. On the most important of these cases, namely, the 

 transmission of motion from one medium to another, vid. Art. 15. 



7. We shall now proceed to integrate the equations (N), for the particu- 

 lar case of plane waves and rectilinear vibrations in a homogeneous body. 



Assume, 



^ = cos If (ax + by + cz — vt), 



9/ =: cos m.f (ax + by + cz — vt), 



f = cos 71. f (ax + by + cz — vt) ; 



where a, b, c are the cosines of the angles which the wave normal makes witli 

 the axes, and /, m, n are the angles made by the direction of vibration. 

 Substituting these values in (N), we find 



— ev^ cos ^ = III cos / + <l>i cos W + ^i COS TO, 



— et/^cosm = 112 cos / + <I>2 cos m + ^I'j cos n, (0) 



— ev'^ cos n = ris cos Z + <l>3 cos m + •^'3 cos n ; 

 where 



n, = ^„.„, a' + A^.^, 6- + ^1,=„- c" + 2y1^,„. he + 2^„,„. ac + 2 J„^., ab, 



*, = B^.^. a' + P^.,- b' + P,,,. c- + 2P,3,v be + 2P„,„. ac + 2P,^„. ab, (U') 



M^, = C„,^, a- + C.>„. b- + C,.„, r + 2C'^,„. be + iCl,^. ac + 26'„^„. a^- ; 



the values of n^, <I>2 , ^.^ , Ilj, O3, ^^ being deduced from those of n, , <1>, , >!/■,, 



by changing, as before, a into /3' and 7'. Assuming 



s = — ev-y 



