IN CRYSTALLIZED MEDIA. 337 



9. To find the last term of the equation 



ft /jc-[/i-3(m cos' + psm-0)\ {/i-3(m cos^ <p + pshr cp)} \k-3{mcos'^l^+psm'\l^)\ 



= 2h' + 9h \{m - p)Y+ 2mp +;/] 



- 27 ! m'g + ji (/- g) + vfp (/- 3g) + mp' (1 - •if+ 3g) \ 



= - 2 {m + 2pf + 9 {m + 2^j) [{m - pff+l^^ + Z»ip\ 



- 27 1 {ni - pfg + p (jn - pfj'+ mp" \ ; 



aX' + h Y' + cZ' = 9 {m - pY\ {»i + 2p)f - 3 {3mg + p (/ - 3^^') \ 



= 9{m-pY\{ni-p).(/- 9g)\ 



= 9{m-jynf-9g); 



2XVZ= 54 (m -lif.g; 



:. - abc + aX' + hV + cZ' + 2XYZ 

 = 2 (»« + ^pj — 9 {ni + 2p) (jf + 2mp) + 27 mp^ 

 + 27 (m - pf \ (m - pi).g +pf]-9 {m + 2p) {m - pff 

 + 9 {m - pf (/- 9^) + 54 {m - jjf -g 

 = {m + 2p) (2;«^ - lOmp —p") + 2Tmp^ 

 = 2m' — Qm'p + 6mjf — 2p^ 

 = 2 (m - pf, 

 which is also independent of the direction of vibration. 



10. By the substitution of these values the equation in A is 

 reduced to the simple form 



A'' - 3 (m - jiy A + 2{m — ji'f = 0, 

 the roots of which are evidently 



m — J), m — p and - 2{m - p). 



This result shews that the vibrations are transmitted with the same 

 velocity in every direction in which such can possibly be transmitted. 



