460 Mr green, on WAVES IN A VARIABLE CANAL 



' = /J + •^" + ^" 



Again, the condition (2) gives by equating separately the coefficients 

 of powers and products of y and s, 



(2'). 



= &c. 



If now by means of (a'), (6'). (c) we eliminate <p" <p„ from (2'), there 

 results 



^*'- ^~ daf ^ \l3dx ydx] dx gy \dfl' 

 It now only remains to integrate this equation. 



For this we shall suppose /3 and y functions of x which vary very 

 slowly, so that if written in their proper form we should have 



/3 = ^{wx), 



where w is a very small quantity. Then, 



d& , , / , d'ji „ 



-i- = w^ \wX), -j—i = <" T ("'^)' °^^- 



Hence if we allow ourselves to omit quantities of the order (o% and 

 assume, to satisfy (4), 



</,„ = Af{t + X\ 



where ^ is a function of x of the same kind as /3 and 7, we have, 

 omittmg y-^), 



'dF~^-^ ' 



d^o _ J dX f, dA 

 dx~ dx'' dx-'' 



d'(b, , (dXy .„ . d'X .,^^dA dX „ 



