Change of Form of Long Waves. 423 



make an exception*; but even Lord Rayleigh and McCowan t, 

 who have successfully and thoroughly treated the theory of 

 this wave, do not directly contradict the statement in question. 

 They are, as it seems to us, inclined to the opinion that the 

 solitary wave is only stationary to a certain approximation. 



It is the desire to settle this question definitively which has 

 led us into the somewhat tedious calculations which are to be 

 found at the end of our paper. We believe, indeed, that from 

 them the conclusion may be drawn, that in a frictionless liquid 

 there may exist absolutely stationary waves and that the form 

 of their surface and the motion of the liquid below it may be 

 expressed by means of rapidly convergent series. But, in 

 order that these lengthy calculations might not obscure other 

 results, which were obtained in a less elaborate way, we have 

 postponed them to the last part of our- paper. 



First, then, we investigate the deformation of a system 

 of waves of arbitrary shape but moving in one direction only, 

 L e. we consider one of the two systems of waves, starting in 

 opposite directions in consequence of any disturbance, after 

 their complete separation from each other. By adding to the 

 motion of the fluid a uniform motion with velocity equal 

 and opposite to the velocity of propagation of the waves, we 

 may reduce the surface of such a system to approximate, but 

 not perfect, rest. 



If, then, l + rj {rj being a small quantity) represent the 

 elevation of the surface above the bottom at a horizontal dis- 

 tance x from the origin of coordinates, we have succeeded in 

 deducing the equation 



<*9 = 3 fg _ ^{iv 2 + §"V + *°-||) 



d*?_ 3 / 



~dt " " 2 V 



where a is a small but arbitrary constant, which is in close 

 connexion with the exact velocity of the uniform motion given 



TZ 

 to the liquid, and where <r=^l 3 depends upon the depth 



ft/ 



I of the liquid, upon the capillary tension T at its surface, and 

 upon its density p. 



On assuming ^- =0 we of course obtain the differential 

 ot 



* Though the theory of the solitary wave is duly discussed in the 

 treatise of Basset, the inconsistency of his result with the doctrine of 

 the necessary change of form of long* waves seems not to have sufficiently 

 attracted the attention of the author. 



t Phil. Mag. 1891, 5th series, vol. xxxii. 



2F2 



