Highest Waves in Water. 433 



the constant being so chosen as to make things right at 

 infinity. 



Inteorating, we have 



U=:log(— ismwyte-X* \-a f + ia l e 2iw + ... , 



al, since U is real at ^=oc 

 >rm 



logA(l + c 1 e 2, ' M ' + c 2 ^'' M ' + ...), 



where a f is real, since U is real at yJr=co ; or, writing the 

 series in the form 



we have 



~ = A(— isin w)-*e~$ iw (l +c l e* iw + c 2 e 4iw + . . .), 



the velocity at yfr = co , or the wave- velocity, being 2~~*/A, 

 and the real root of (— isiniv)~* being taken along </> = 0. 



Suppose the units so chosen that A=l, and hence V=l/2*, 

 and the length of wave L = 7r/V — it x 2*. 



dz 

 It will be more convenient, at first, to invert -=— and write 



7 dw 



dw , , 



-j£ = ( — i sin wf e&«( 1 + ^ e 2iw + b 2 e* iw + ...)• 



On the surface between = and = 7r, 



J = sint £*K*-f )(l + j^ + j^ + . . .) ? 



the real root of sin* being taken. The surface-condition 



may be written 



or 



dcj> y dy 

 Taking 



+ (263 + 26^2)^ + ...}, 



and multiplying by its conjugate, we get, omitting terms of 

 order 4 in the &'s, 



2 4 =sin^>{l-f4V+ (4&! + 8& 1 & 2 + 46 1 3 )cos 20 



+ (46 2 + 2 V) cos 4 + (4Z> 3 + 4&! fc 2 ) cos 60 } , 

 PM. 3%. S. 5. Vol. 36. No. 222. JVot?. 1893. 2 G 



