1885.] third order in an isotropic elastic medium. 299 



appears to be no ground for supposing that the above relation 

 however would hold for every medium through which transverse 

 waves might be propagated, and the retention of the above cubic 

 terms leads to interesting results, which we shall now consider. 



(6) It is possible for a plane wave, given by equations (ii), to 

 remain in a certain sense polarised in one plane. For it is only 

 needful to put v = 0, which is a possible solution of the second 

 equation, and the first then involving only u becomes 



v du z 3 2 d 2 u _ d 2 u 

 S~aV + K dz~ 2 ~dt 2 ' 

 du\ 2 d 2 u „ d 2 u d 2 u 



\dz) dz 2 dz 2 ~ df ' 



As a first approximation let us take a single term of the type 



2tt / 

 A cos -z-(z — Kt). 

 A 



Then 

 [dt 2 - K d?) U = - A ^^rn-(*-**)cos T (s-«*) 



.„ 1677- 4 27T , ^ 



= — A 6 — j- v cos — (Z — Kt) 

 A A. 



.0 l07r q 27T , . 



+ A* —j- v cos 3 — - (z - Kt) 

 A A 



.0 47T* 2-7T N ^473"* 67r. 



= — -4 -r^- y cos— - {z — Kt) +A —j- J» COS — - {Z — Kt). 



A A A A 



Hence u = A cos — (z — Kt) + A s -j - t sin — iz — Kt) 

 A A k A 



., 7T 3 y , . 67T , ,, 



— A*—;=-tsm—(z — Kt). 

 \ 3 Sk A v ' 



These results are of a rather remarkable character, the ad- 

 ditional terms introduced contain Us a factor of the amplitude, 

 and although these may only be the first terms of a series of powers 

 of t which may not necessarily become infinite with the time, yet it 

 would seem that a wave of velocity k could not be propagated in 

 the medium as a stable motion. There would arise superposed 

 waves of increasing amplitudes of (i) the same wave length but 

 retarded by a quarter of a wave length, (ii) of one-third the wave 

 length. 



Has anything of this kind been observed ? It would suggest 

 that if a chemical ray were selected and passed through a proper 



