1 68 On a Dynamical Theory of 



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



- 8V =X' ( j - - ) 4 FV[ 7-^ r~^ ) + Z'o ( r^- r^ ) 

 \ az ay Q j \ dx dz^ J \ dy dx J 



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

 , y, s, 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 " , /", " . Denoting the cosines of the angles which 

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

 with the axis of y by /', m', n' ; with the axis of z by I", m", ri' ; 

 and putting 



dx dz ' dy dx ' 



T^yrr j^fff i^fff j^ " 



doc, o dec, o ^r?,; oc o orj o 



-. : : , Q : : , 



dz cM/o dx dz 



we have 



These expressions for SX, Y, %Z 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. 



