Doctrine of Extraneous Force. 125 



[2(2G + L-M-N)X 2 -L]^ + [2(2H + M-N-LV-M]?/ 2 



+ [2(2I + N-L-M> 2 -N> 2 = const. . (30) 



is cut by the plane 



\x+jny + vz = (31). 



If the displacement is in either of the two directions (I, m, n) 

 thus determined, the force required to maintain it is in the 

 direction of displacement ; and the magnitude of this force 

 per unit bulk of the material of the plate at any point within 

 it is easily proved to be 



wg ( 32 )> 



where {M} denotes the maximum or the minimum value of 

 the bracketed factor of (29). 



18. Passing now from equilibrium to motion, we see at 

 once that (the density being taken as unity) 



Y 2 ={M} (33), 



where V denotes the velocity of either of two simple waves 

 whose wave-front is perpendicular to (X, /i, v). Consider the 

 case of wave-front perpendicular to one of the three principal 

 planes ; (vz) for instance : we have X=0 ; and, to make {} 

 of (29) a maximum or minimum, we see by symmetry that 

 we must either have 



(vibration perpendicular to principal plane) 1=1, m= 0, n = 0-\ 



(vibration in principal plane) 1 = 0, m= —v, n=[MJ 



Hence, for the two cases, we have respectively : 



Vibration perpendicular to yz . . . V 2 = M + N + (B — M)yu, 2 + (C — N)j/ 2 (35) . 



Vibration in yz V 2 = L + B/i 2 + CV + ^H + I-L^V (36). 



19. According to Fresnel's theory (35) must be constant, 

 and the last term of (36) must vanish. These and the corre- 

 sponding conclusions relatively to the other two principal 

 planes are satisfied if, and require that, 



A_L:=B-M = C-N (37), 



and 



H + I = L; I + G=M; G + H = N . (38). 



Transposing M and N in the last of equations (37), substituting 

 for them their values by (23), and dividing each member by 



