233-234] THE SOLITARY WAVE. 419 



present, where one of the family of stream-lines is straight. We derive 

 from (i) 



where the accents denote differentiations with respect to x. The stream-line 

 ^ = here forms the bed of the canal, whilst at the free surface we have 

 \fr = - cA, where c is the uniform velocity, and h the depth, in the parts of the 

 fluid at a distance from the wave, whether in front or behind. 



The condition of uniform pressure along the free surface gives 



u* + v* = c*-2g(y-k) .............................. (iii), 



or, substituting from (ii), 



F"t-y*F'F'" + y*F"* + ...=:c*-<2g(y-h') ............... (iv). 



But, from (ii), we have, along the same surface, 



yF'-^F'" + ...= -ch ............................. (v). 



It remains to eliminate /^between (iv) and (v) ; the result will be a differential 

 equation to determine the ordinate y of the free surface. If (as we will 

 suppose) the function F' (x) and its differential coefficients vary slowly with 

 x, so that they change only by a small fraction of their values when x increases 

 by an amount comparable with the depth k, the terms in (iv) and (v) will be of 

 gradually diminishing magnitude, and the elimination in question can be 

 carried out by a process of successive approximation. 



Thus, from (v), 



and if we retain only terms up to the order last written, the equation (iv) 

 becomes 



or, on reduction, 



q o -.9 19 -919 ..\T**/ 



t7 



If we multiply by y', and integrate, determining the arbitrary constant so 

 as to make y' = Q for y =A, we obtain 



~^ + 3~y = ~A + ~P~ 

 or y 2 = 3^^- 2 (l-*|) (viii). 



Hence y' vanishes only for y h and y = c 2 /#, and since the last factor must 

 be positive, it appears that &\g is a maximum value of y. Hence the wave is 



272 



