22B ON WAVES IN A VARIABLE CANAL 



If now by means of (a'), (&'), (c) we eliminate $"</> from 

 (2'), there results 



_, , . 



~ dx 2 + \J3das + ydx) dx gy\ df ) " 

 It now only remains to integrate this equation. 



For this we shall suppose ft and 7 functions of x which 

 vary very slowly, so that if written in their proper form we 

 should have 



= ^(a>x), 



where w is a very small quantity. Then, 



'd& ,,, v d*ft ,., . - 

 ^ = co^fr (o)^), -j-g = eo 2 ^ (cox),- &c. 



Hence if we allow ourselves to omit quantities of the order 

 a) 2 , and assume, to satisfy (4), 



where A is a function of x of the same kind as ft and 7, we have, 

 omitting (g), " ;: '' V " ; 1, l 



^ 



t 

 dx dx j 



Substituting these in (4), and still neglecting, quantities of the 

 order eo 2 , we get 



91 



*AdX (dp 

 dx -fa + d 



