310 Mr. E. F. Gwyther on the 



The solution of this equation would give the general motion 

 of a progressive long wave, and would allow us to determine 



the permissible values of fi. After putting/' = — — 4 — , 



we obtain an equation which would perhaps be most conve- 

 niently treated by Weierstrass's notation, since the factors on 

 the right-hand side are far from obvious. We should be 

 entitled to put c^—gh in the coefficients of the small terms in 

 order to make the expressions simpler. If d 2 =gh exactly, the 

 constant would have to be treated differently. As, however, 

 my present object is to examine the nature of the approxi- 

 mation which has led to the differential equation, I shall 

 confine the further integration to the case where fi = 0, which 

 leads to the case of the solitary wave. In the case of this 

 wave we are able not only to compare our results with the 

 analysis of earlier mathematical investigations, but also with 

 the records of Scott Russell's careful observations. 

 Putting fjt, = in (15) we obtain 



8 (,_£)*./*. ( 4+ J. ( ' v -^g- g/,) )^- y %" 



/ 2 (5c 2 -gh)(c*-gh) \ 



"A3" ^_9hY r-' 



3/ 

 + 4/'\ 



The first approximation being /' = — sech 2 mx } where 



m 2 h 2 =(c 2 -^gh) l2fc 2 —~\ we are guided to the form of the 



second approximation, namely, 



where 



and 



j,f_ sech 2 mx . 



J ~^l-^tanh 2 »^' W 



c' z — ah 

 9= — 5T- 



