VIBRATORY MOTION OF AN ELASTIC MEDIUM. 521 



If we substitute in (8) the values of the Cs given in (9), we find the following value of U", viz., 

 A, 



dydss 

 and therefore (8) becomes 



d (dr, dr\ d Idt d^\ d tdl dr,\ 



dx \dy dzj dy \dz dxi d« \dw dyl 



d«dx 



dxdy 



d-« / d d d\ (d^ dr, d^ 



dt^ \ dx dy dmi \dx dy dx 



^ I d ^ d \df „, I d d \dP 



\ dy dxj dy \ dz dx/ dz 



^ I ^ d d\ dri „ , I „ d d\ dri 



+ 5J/3 — -7— --' + S.' /3- -a-]-r^ 



V dz ' dy) dz V dx dyl dx 



d 



.(10). 



+ BAy—-a — ]-f + B, U,— -/3— -f 



\ dx dzj dx \ dy dx) dy , 



d n d r.—, d d 



By using the notation in Art. 8, &c., and observing that « "t - p-r—=Dm.y, a— - 7^ 



= — DlB • (3, &c. &c. the equation (lO) becomes 



d ^ d d \ ^ 



df- r'"Xr ^^■'^'dy ■ --- dzl- 



+ DW 



{{"''£-'>■?,)'* {''■?^-''r:)'* {"''4 -'•i^)-'] 



,(11). 



For transverse vibrations we have AI3 . « = O, and therefore, 



df 



= Z)D 



Now Du ,u is the symbol of a line perpendicular to u and m; hence (12) indicates that the 



cPv 

 force -— is perpendicular to the direction of W, and that direction, as we have seen, is the direc- 

 tion of propagation. It follows, therefore, that if the relations (9) hold, the forces brought into 

 play by transverse vibrations are always perpendicular to the direction of propagation. 



25. AVe shall now shew that this cannot be the case except the conditions (9) hold. 

 If the conditions (9) do not hold we must add to the second member of (12) an expression of 

 the form 



^(Z:,,7 H- E^m + rf^(^^r« -^ El^y) + ^(^»e/3 -»- E.;.,.) = V, suppose 



E, El Ei EJ E-i Ej being the unknown corrections to be made in the second members of the 

 relations (.9). 



Performing on V the operation AH)., we find, 



dydz\ dz dy) dxdw\ dx dzl dxdy\ dy dx) 



Now, if the second member of (12) + V is perpendicular to the direction of TB, the same must 



S \2 



