250 Prof. C. Niven on the Theory of an 



duction of Green's theory, and is somewhat more easily applied 

 than his process or Green's. Let the small motions of a solid 

 be given by the equations 



d 2 u ,. ( d d d^ s d 2 v j. / d ^ N 



then, if there be two transversal waves in every direction, the 



„ d?6 d 2 (du , dv , div\ ,. „ ,, „ 



expression lor -^ or ~n, \ ~T + T~ + ~T~ ) mus t be or the form 



dy 



*(t>7T>t) ' 



\ax dy dz/ 



dy 



For let us consider, first of all, an ordinary homogeneous 

 solid, for which f 1} f 2 , / 3 are quadratic functions of ■=-? -=-* -=-'. 



iXJL CLLf CL£> 



We may satisfy the equations by putting u,v, w = (u ,v ,w ) sin f, 



2tt 

 where J == — (Ix + my + nz—Yt), obtaining the three following 



A. 



equations, 



u , v , w . V 2 =/i, f 2 , / 8 . (Z, m, w Jm , v , w ). 



But if V 2 (?« Z + f m + zt' n), derived from these, do not con- 

 tain u l + v m + w n as a factor, we shall have two equations of 

 the form u Q l + v m + w n = 0, w Z' + v m / + w Q n f = 0, giving a 

 single value for the ratios u : v : iv oy and by consequence a 

 single value of V 2 , contrary to the hypothesis. 



The equations of motion of the solid take the form, as Lame 

 has shown, 



W- Zb ''dz~~ lG ^ 

 dr l doc 1 dz 



wherein a ly 6 1? c x are the three principal wave-velocities. 

 In the case where f 1} / 2 , / 3 contain third as well as second 



powers of -T- . . . , the necessary condition is clearly fulfilled by 



supposing the principle stated to apply to both sets of terms; 

 and it may be readily shown that this is the only conclusion 

 which will satisfy it. 



Reduction of terms of Class (I., III.). 

 9. The part of the energy arising from this source will be 



