1 68 On a Dynamical Theory of 



and suppose the velocity of propagation to be unity, we have 



z dy Q ) \ dx Q dz 



For the crystallized medium, if its principal axes be those of 

 a?, y, a, the value of 8V" will be the same as that of 8V in for- 

 mula (3) ; but instead of the variations of ?, rj, , we must use 

 those of " , ij" , " . Denoting the cosines of the angles which 

 the principal axes respectively make with the axis of X Q by /, m, n ; 

 with the axis of y by f , m', ri ; with the axis of Z Q by T, m", n" ; 

 and putting 



Yf, _ dvi"o d%" _ dZ" d" _ d" Q dr{' 9 



"**- ~ ~~j i > * o~ ; ; > " o j j > 



azo dy Q dx Q az dy dx Q 



we have 



= 



These expressions for SX, 8 F, SZ" having been written in 

 formula (3), the resulting value of 8 V", as well as the above value 

 of 8V', is to be substituted in the equation (17), and then the 

 right-hand member of that equation is to be integrated by parts, 

 in order to get rid of the differential coefficients of the varia- 

 tions. When this operation is performed, the triple integrals 

 on one side of the equation will be equal to those on the other ; 

 and by equating the coefficients of the corresponding variations 



* It is assumed here, and in what follows, that when there are two or more 

 coexisting waves in a given medium, the form of the function V is the same as for 

 a single wave, provided the displacements which enter into the function be the re- 

 sultants of the displacements due to each wave separately. This, however, ought 

 evidently to be the case, in order that the principle of the superposition of vibrations 

 may hold good. 



