APPENDIX. 335 



Note to pp. 301, 302. 



These results may be otherwise obtained by the consideration 

 that if one of the waves be propagated by normal vibrations, the 

 corresponding values of a, ft 7 must be proportional to a, b y c. 

 We thus obtain, from equations (6), 



A' a + F'b + E'c Fa, + B'b + D'c E'a + D'b + C'c 



= 1 = = P e 



a b c 



Now replacing A, B, G', D' ', Ef, F' by their values, we see 



. . A a -f Fb + E'c . , A 



that is equal to 



a 



or Go* + (2JV+ R) V + (2M+ Q)c 2 + Aa* + Bb* + Cc\ 



The second and third members of the above equation being 

 similarly transformed, we see that 



Ga*+ (2N+ R) b 2 + (2M+ Q) c 2 

 = Hb 2 + (2L + P) c* + (2AT+ E] d* 

 = /c 2 + (2M+ Q) a* + (2L + P) c 2 , 



for all values of a, 5, c ; which leads at once to the equations 

 given at the foot of p. 302 ; and also proves that the normal 



, .. , ... , /A*- Aa? - Bb* - Cc*\l 

 velocity of propagation e is equal to (- 1 , 



which, when the system is free from extraneous pressure, be- 



/ u, 

 comes 



If the values of A, B r , C', D' y E', F' in terms of /*, L, M, N 

 be substituted in equations (6), we obtain the following equation 

 for the determination of pe 2 , the system being supposed free from 

 extraneous pressure : 



fjia*+Nb*+Mc*-pe\ (p-N) ab, (/* - M) ca 



-N) ab, fjLb*+Lc*+Na*-pe 2 , (/* - L) be 

 - M) ca, (jA-L) be, 



= 0, 



