50 Mr. J. M c Cowan on the Solitary Wave. 



the form of the solitary wave is given by the equations (11), 

 (17), and (18), and its velocity of propagation by (16). 

 We may from these eliminate, when required, any two of the 

 three constants a, rj 0: and m, but it is in general convenient to 

 retain them, as each has a direct physical significance. 



For purposes of approximation it should be noticed that 

 when mh is regarded as a small quantity of the first order, 

 then, by (17) and (18), ma will be of the second, and mr] of 

 the third order. We proceed to consider certain rough 

 approximations. 



If in (11) we neglect m 2 h 2 in expanding the cosine, &c, it 

 reduces by means of (18) to 



7) = 7j sech 2 ^ma;, (21) 



which is the approximation to the surface obtained by Boussi- 

 nesq, and again by Lord Rayleigh. 



Similarly (17) and (18) give for a first approximation 



m % =im 3 /i 3 orm=v/(3 % /A 3 ), . . . (22) 



as found by Boussinesq : Rayleigh obtained 



m = V { or) /h 2 ( h +%)}., 



which is a little nearer, for, proceeding to the next approxima- 

 tion, (17) and (18) give 



m=^{3 V o/h 2 (h+\? 2 - Vo )\ (23) 



Treating (16) similarly we obtain 



W=gh(l + lm 2 h 2 ) 

 or 



W=g(h + % ), (24) 



the approximation obtained by Boussinesq and Rayleigh, and 

 the result originally deduced experimentally by Scott Russell 

 and confirmed by Bazin*. It is, however, to be noticed that 

 the experiments of Russell and Bazin cannot be regarded as 

 capable of discriminating between the approximation of (24) 

 and the more exact result given by (16). This will be 

 sufficiently obvious to those who have had experience of such 

 measurements, and it need only be pointed out that the 

 experiments on which Russell relied to establish (24) were 

 made in a long trough 20 or 30 feet long, and that, so far as I 

 am aware, no allowance was made, nor I think could well 

 have been made, for the influence of the successive reflexions 

 from the ends. 



* Mem. des Savants etrangers, torn. xix. 



