1885.] third order in an isotropic elastic medium. 301 



is deduced from the ordinary theory of elasticity and without any 

 assumption as to the ratio of wave length to molecular distance. 

 We remark in the second place that the wave of length A, will 

 be accompanied by a wave of one-third that length, which has the 

 intensity a function of the time. Here again we have only the 

 first term of a series which does not obviously become infinitely 

 great as t increases. If u represented a wave of light (not a chemi- 

 cal ray from beyond the violet) for example, the wave of one-third 

 its length would not fall within the sensible spectrum, and hence 

 there might be some difficulty in ascertaining whether the above 

 anomalous term had a real existence. It can hardly be doubted 

 however that if it does exist it ought to manifest itself in some 

 manner, for it would seem to correspond to the breaking up in 

 some fashion of a ray of light of a single wave length transmitted 

 through an isotropic medium. An isotropic medium would seem 

 in a certain sense to possess 'double refraction' for a selected ray, 

 it divides the ray into two parts, one of which has one-third of the 

 previous wave length. 



7. If we take u of the most general form 



= XA n cosn{z-k n t), 

 the anomalous terms do not necessarily appear unless one of the 

 k n 's = k. The treatment of this general form involves considerable 

 difficulties, which I postpone for the present, as it requires careful 

 examination, being not unsuggestive for the problems of absorp- 

 tion. 



8. (a) Let us return to equations (ii) and suppose v is not 

 zero, but that to begin with 



2tt 2-7T 



u = A cos — - [z — id), v = B cos — - (z — ict). 

 We find to the second approximation 

 u = A cos 2 ^ (z - K t)+ A (A 2 + B z )^-t sin— (z-Kt) 



A. A. K \ 



-A(A 2 + B 2 ) % z £ t sin ^ - K t), 



A, OK A 



v = B cos — {z-Kt)+B (A* + B 2 )'£-tsm 2 ^(z-Kt) 



Aj a, k A. 



T) / A 2 n»\ IT V , . 67T . . 



- B (A 2 + B-) ^—t sm — (z - K t), 



which corresponds of course to what we have considered above, i.e. 

 a plane wave of vibrations all parallel to a fixed direction, in this 



case tan -1 -j and of intensity A 2 + B 2 . 



vol. v. pt. iv. 21 



