Change of Form of Long Waves. 441 



Proceeding to the second approximation, we find 



and then again from (51) and (52), 



o #/i , h — k\ 3 15 A — /: /rK > 



Finally, a third approximation leads to : — 



qj h-k 9 (h-k) 2 93M\ : 3 M 



^~ ZV + 2/ 20 / 2 80Z 2 j ; ^ + 49 *Z 2 ; 



^ 2 £ /W 21 </ (A-£) 2 12 g M 

 f Z * Z 20/ Z 2 5 / ' Z 2 ; 



o 7/1 . ^ — ^ 1 (h — ky 33 hk\ /K > 



2 2 =n i+ -7 — 20V 2 -2of)- ■• • • ^ 



By means of these results we may now readily obtain from 

 (1) and (2) expressions for « and v including respectively 

 the terms of the 2 nd and 2J th order. 



They are : — 



' J"/, , h-k 3 {h-kf 33/*A\ , /, , h-k\ v 



"=Vf{( 1+ ^-i-?> + (-¥ + i-r> 3 } ; ^ 



where 



^=|(l-|^)(A-,)(i+,)(l + |.|). . (59) 



VI. Calculation of the Equation of the Surface. 



We will now show how for the equation of the surface of a 

 stationary train of waves a more correct expression than (20) 

 can be deduced. For this purpose we have to integrate 

 the differential equation (39), or rather we have to prove that 

 a series can be given which solves this equation to any desired 



