1885.] third order in an isotropic elastic medium. 303 



order (amplitude) 5 the vibration remains circular although the 

 radius then alters with the time. It must be noted however that 

 there would be in this case superposed anomalous waves, for we 

 should have to find u and v to a third approximation. 



These results seem noteworthy, for it would appear that a 

 wave of circular vibration is more stable than a wave of linear 

 vibration, and that whereas a wave of linear vibration is accom- 

 panied by anomalous waves which rapidly rise into importance, 

 these disappear in the case of a wave of circular vibration to a 

 much higher degree of approximation. Further it would seem 

 that, disregarding the anomalous wave, elliptic vibration with a 

 slow uniform rotation of the ellipse may for a very considerable 

 time be a stable form of wave motion in an isotropic elastic medium. 

 These remarks again have interest for the undulatory theory of 

 light. 



9. Let us assume more generally that 



u = A cos (mz — nt), v — B cos (mz — nt + a), 



Then we find : 

 ^ _ k^ = - j£-A (3m 2 ^ 2 + 4>p*B 2 ) cos {mz - nt) 



m*v . , _ . ,. 



H — — A cos 3 (mz — nt) 

 4 



m(m + 2p)p 2 v ™ , , — — _ . 



+ — 19 B 2 A cos (m + 2pz -n + 2qt + 2a) 



, on(m — 2p)p i v n2 , . — - — 



H 19 ^ ^ cos (m - 2pz -n-2qt- 2a), 



whence it follows that 



a / *\ m*vA s . 1 _ . ,. 



u — A cos (mz — nt) tt. — t sm S (mz — nt) 



v ' 24w v ' 



m (m + 2p) p^vB^A 



24 (n + 2g) r a ' 



on (m — 2p) p i vB 2 A 



aA , ' , £ sin (m — 2pz- n — 2qt— 2a), 



24 (n —2q) v y * /' 



, , 2 2 ?rcM 2 mW 



where n" — kW = - - H ^ — 



4 b 



21—2 



