364 Prof. Stokes, On highest wave of uniform propagation. [May 14, 



from choosing the direction of x positive as that of the propagation 

 of the waves in the first instance, and therefore the direction of x 

 negative as that of the superposed velocity by which the motion is 

 converted into steady motion. 



x having been determined in terms of <£, x + $ is to be deemed 

 a known function of <£,/(</)) suppose, which will have 2ir for its 

 period. The coefficients A n may then be deemed known, and on 

 substituting in (2) we shall have y expressed in terms of /(</>)• 

 Denoting this expression for y by F (<j>) we shall have 



F(cf>)=-tj^ &nk + e _ nk cos ncf> sin wf / (f ) d<f>', ... (3) 



where of course the integration with respect to <£' may stop at ir if 

 we double the coefficient, since /(27r— </>) =—f(<f>). 



If the trial curve had been the true outline, the curve of which 

 the ordinate is determined by (3) would have come out identical 

 with the original, which would have proved the original to have 

 been correct : otherwise the new curve will be a much closer 

 approximation to the true form than the trial curve, and may be 

 used as a fresh trial curve, and so on. 



In (3) the integration is supposed to be performed first and 

 then the summation. If we attempted to perform the summation 

 first, we should encounter a series which is neither convergent nor 

 divergent, but fluctuating. Such a series may however be summed 

 by regarding it as the limit of the convergent series formed from it 

 by multiplying the ?ith term by the ?*th power of a quantity less 

 than 1, and which is supposed to become equal to 1 in the limit. 

 The summation cannot however, so far as I know, be actually 

 effected except in two cases. 



The first case is that of a fluid of infinite depth, for which the 

 fraction involving the exponentials becomes equal to 1, and the 

 series divides into a pair of series of sines of arcs in arithmetical 

 progression, which may be summed by regarding it as the limit of 

 another series ; a view to which we are naturally conducted by 

 regarding the stream line of the surface as the limit of a stream 

 line taken first a little below the surface. The other case is that 

 in which the wave length is regarded as infinitely great instead of 

 infinitely small compared with the depth of the fluid. In this case 

 we first take a crest for the origin of x and then make A, infinite, 

 when the sum takes the form of a definite integral which may be 

 evaluated according to the known formula 



e ax _ e -ax _ ^U 



sin oxax = 



oax i o~ax QTrbi 'In Q~™b ~" ' 



.As the other crests have moved off to infinity, we arc in this 

 case left with a solitary wave. 



