1885.] third order in an isotropic elastic medium. 305 



the same wave lengths as the principal waves introduced into 

 their respective vibratory planes. 



(iv) If the two principal waves have equal wave lengths, they 

 will as a rule have different velocities of propagation, depending 

 not only on their own, but on each others' intensities. The 

 velocity of a wave in one plane is accelerated by the existence 

 of a wave in a plane at right angles. 



(v) Equal wave lengths and equal velocities of propagation 

 necessitate equal amplitudes, but in this case there will be ano- 

 malous waves of the same wave length and velocity arising from 

 the terms with the phases 



m — 2pz — n — 2qt — 2% 

 ! p — 2mz — q — 2nt + a) 



The above results might all be translated into theorems con- 

 cerning plane polarised light, but it seems idle to restate in the 

 language of optics theorems which may after all have no bearing 

 upon that subject. 



10. We have in the course of our work (Art. 3) supposed 

 that w — 0, or that there is no wave of normal vibration. It may 

 be as well to inquire a little more fully into the legitimacy of this 

 supposition. If w be not zero we must take 



where <r = w z and the other strains remain unchanged (Art. 2). 



Now the expression we obtained for the work in Art. 4 may 

 be written 



2TF = /z(a 2 + /3) 2 4-^(a 2 +/3 2 ) 2 , 



, X + 2/Jb 4- 4 (c + a) 



where v = : — , 



4/> 



if we substitute for s 3 in that expression |(a 2 +/3 2 ). 



If a be not zero we must take for W, 



2W = fi (a 2 + /3 2 ) + ^ (a 2 + /3 2 ) 2 + (X 4- 2/*) a 2 + ecr s +fv* 



+ (\ + 2fi + 2c) a- (a 2 + yS 2 ) + ha 2 (a 2 + /3 2 ), 



where c is the same constant as before. We may write 



\ + 2/i + 2c = 2e, 

 dW 

 ■'• ~da = (X +-ri a + ¥<** + 2 /°- 3 + e (*" + £*) + A* (« 2 + P)> 



