1885.] third order in an isotropic elastic medium. 



297 



compressional wave ; then u, v will be functions of z only, and iv 

 will equal zero. Hence it follows that 



s 1 = s 2 = y = 0, *.= i(«. 1 +0» 



a = v z , fi = u z , 



W = F(a,/3,s 3 ). 

 Lagrange's method gives us at once : 



r dhi ~ d 2 v 

 ^ 8u + d? 

 to determine the vibrations. Or : — 



= f[[8Wdxdydz + p 



&u + -77^ Sw ) dxdydz 





= 





+ P 



S!!{w Su+ aw Bv ) dxd ^ dz - 



The first integral may be written 



dW( dSu . d8v\ dWdSv dW dSu \ 

 ds 3 { z dz " dz) da. dz djd dz j 



8u- 



+ 



doL 



dxdydz 



Integrating by parts and neglecting the surface terms, we find 



d fdW dW\ 8v ±(dW v dWy 

 dz\ds 3 z d/3 ) dz\ds 3 



or the body equations take the form: 



d fdW dW\ dhi^ 



dz \ ds, c 



d fdW 



dz \ ds. 



v. 4- 



d/3 J 



dW \ 



doc J 



dt 



ay 



df j 



(i). 



4. We must now determine the form of W. So far as terms 

 of the second order are concerned W must be of the form 



2W=\s 3 2 + f x(a 2 + ff i + 2s 3 i ). 

 It is needful however to consider possible cubic and certain quartic 

 terms. The only terms of the 3rd and 4th order in the expression 

 for the work which would not give rise to terms higher than the 

 third in the differential equations are of the form 

 cs 3 (a 2 + /3 2 ) + ds 3 OL0 + e (a 3 + /3 3 ) +faj3 (a + 0) 



+ g (a 4 + /3 4 ) + ha 2 2 + i (a 2 + 2 ) a/3. 



If we turn the axes through any small angle 86, s 3 remains un- 

 changed, a becomes a — 086, and ft, + a80. Hence, since this 

 cannot change the form of W it easily follows that d, e, f are all 

 zero and h = 2g. 



