■("), 



298 Prof. K. Pearson, On plane waves of the [May 25, 



Thus W is of the form : 



2 W = H 2 + A* (* 2 + /3 2 + 2s 3 2 ) + 2c5 3 (a 2 + /3 2 ) + # (a 2 + /3 2 ) 2 . 

 We acccordingly find 



d ^=(X + 2 f ,)s 3 +c(a* + n 



dW 



~ = fia + 2cs 3 a + 2ga (a 2 + /3 2 ), 



d ^ = fJ ,(3 + 2cs 3 (3+2g/3(a 2 + F). 



Hence, substituting in (i) we have 



v d r/ „ , 2X -, 2 cZ 2 w d 2 it 



where /e 2 = - and ^ = — r - — — . 



p 3 2/3 



5. We proceed to draw some inferences from these equations. 

 (a) A ' plane polarised ' wave or one of the type 



A 27T . . _, 27T . ,. 



u = A cos — (z — Kt), v = B cos -r- (s — /c£), 



is not an accurate representation of a possible wave motion in an 

 elastic medium unless v — 0. 



This involves A, + 2/u, + 4 (c + #) = 0. 



It seems to me possible that c may equal zero, and in this case 



, \+2a 

 we have g = ^~ > 



or the coefficient of the terms of the fourth order is negative. If 

 we do not look upon c as zero, we have in our particular case, 

 since s 3 = ^ (a 2 + ft*), for the terms of the fourth order 



2W i =(c + g)(a* + /3 2 f, 



and we again find the coefficient of the terms of the fourth order 

 negative. We are thus led to a certain relation between the 

 elastic coefficients of the square and quartic terms of the work, which 

 must be satisfied, if a plane polarised wave is to be propagated 

 through the medium. It is not impossible that such relation holds 

 or very nearly holds for the ether, it would denote that the expres- 

 sion usually taken after Green for the work is too great. There 



