350 Mr. R. F. Gwyther on the 



For this purpose I shall extend to this case the general 

 principles of the method whereby in a late paper* I attempted 

 to deal with the steady motion o£ long waves, by finding the 

 differential equation which the velocity-potential must approxi- 

 mately satisfy. 



§ 1. Taking 



t}) + if = Y(at + iy, t) ..... . (1) 



to represent a state of motion in fluid otherwise at rest, the 

 condition expressing that the pressure is constant along the 

 free surface may be written 



2gh=2gy + u^ + v' 2 -2^, (2). 



where h is an absolute constant. 



Expanding this, and replacing differential coefficients of <f> 

 by their values from (1), and being guided in approximating 

 by the hypothesis that we are dealing with a long wave, we 

 obtain 



2gh = 2gy + F /2 - (F F" - F" % 2 + Ac. 



-2F + FV-&C . . . (3) 



This may be looked upon as the equation to the free sur- 

 face ; but in order that this may continually be the case, we 

 shall have a condition, obtained by operating with 



This condition is 



0=-% + (F"-F'F"' + F"%} |F",y-F iv |^| 



-{2F / F // -2F / + F / V} { F - F '"C} 



+ 2F / F / -2F + F'V + &c (4) 



Substituting in this from (3), for y, and retaining only the 

 most important of the terms under our hypothesis, we finally 

 obtain 



2{ghF"-Y) + 2(FF" + 2FF) -3F /2 F" 



+ F'7^-F iv ^=0. ... (5) 



This is the general equation which I proposed to find, 

 being that which, to a first approximation, the function F 

 must satisfy in any long-wave motion. 



* Phil. Mag, Aug. 1900. 



