VIBRATORY MOTION OF AN ELASTIC MEDIUM. 517 



13. Al^-v, and Di^.v may be found separately by the integration of a differential equation 

 of the form 



d'U f(PU d'U d'U\ 



-d¥ = ^[d^-'lf^^) ^'^- 



For performing the operation A?3 . on both sides of (3), and putting i\?3 .v- U, we find 

 — (AS .v) = B (AS . S) AS .V + (J -B) (AS . S) AS . v ; 



(12 rj 

 or, ^ = ^(AS.S)f7; 



and performing the operation i)S . , and putting Z)S . « = U, we find (observing that DS . S = 0), 



^=fi(Al9.3B) U: 



hence, AiB.i) and D'^.v may be found separately by the integration of an equation of the 

 form (()), C being equal to A in one case, and to B in the other. 



Investigation of the Equation of Vibratory Motion of a Crystnlli%ed Medium. 



14. When the constitution of the vibrating medium is crystalline, we may obtain a differential 

 equation similar to that we have found for an uncrystallized medium, and by exactly the same 

 method ; the only difference will be in the constants introduced, in regard to which we must bear 

 in mind, that the elasticity of the medium is no longer the same in all directions, and therefore the 

 constants A and B, which we may call the coefficients of direct and lateral elasticity respectively, 

 will be different with respect to different directions. The method we have employed to find the 

 force brought into play by a disarrangement of the medium requires us to consider this difference 

 of elasticity only with respect to the three directions a, /3, 7, assuming that the medium is still 

 symmetrical with respect to these three axes, i.e. supposing them to be the axes of elasticity. 



15. Hence, reasoning as in article (5), the force brought into play by the disarrangement, 



^« = ii„g3.^./3^V.7B^^^ 



dy- 



-'"^^ ^-S-^^^^g-^^^-^S' ^^)' 



A^, A-^, A3, being the coefficients of direct elasticity for the three directions a, /3, 7. 



Again, reasoning as in the same article, the force brought into play by the disarrangement, 



will be 



^ •(-■^^-■■S) ^^i^-s*"-^ *^(--s**-i-- <"■>• 



B„ Bi', B.i, J5/, &c., denoting the coefficients of lateral elasticity for the three directions u, /3, 7. 

 We make a difference here between 5. and 5,', because the disarrangements. 



