Lord Rayleigh on Waves. 265 



varies slowly, or (as we may put it) a function of cox, where co 

 is a small quantity. If we agree to neglect the fourth power 

 of co, the third and following terms on the left-hand side of (E) 

 may be omitted, and we obtain 



+ 2 { i +/ 2 - \f ^r } **-?-%?> 



or 



^{i-Jy 2 +|w"}=v-2^ J . . . (f> 



by which the value of ux is determined approximately in terms 

 of the form of the upper surface. If we suppose vr constant 

 and integrate (F) on that hypothesis, we shall obtain a form 

 of upper surface for which the pressure varies very slightly, 

 provided of course that the solution so obtained satisfies the 

 suppositions on which the differential equation (F) is founded. 

 To integrate (F) we may write it in the form 



or 



which becomes a complete differential when multiplied by 



2^-dx. Thus we find 

 ax 



}**-* + =^+1, 



C being the constant of integration. Suppose now that in the 

 undisturbed parts of the canal the depth is I and the velocity 

 u Q . Then 



and 



ir=\u dy=u Q l, 



Substituting these, we get 



In this equation g and I are given, while u and C are at our 

 disposal ; and thus the cubic expression on the right may be 

 made to vanish for y — l and y = V , where V is the distance be- 

 tween the summit of the wave and the bottom of the canal. If 



