1883] of uniform propagation. 363 



theoretical outline of the highest possible wave. But it is tanta- 

 lizing to get thus near it and not to be able to complete the solu- 

 tion. 



The expansion in series would be of little avail for the reason 

 I have mentioned ; but it occurred to me that some method of 

 trial and error might succeed. I have devised one which promises 

 so well that a notice of the method may not be without interest to 

 the Society, though I am not at present in a condition to present 

 the result, except it were in a rough way, not having completed 

 the numerical calculations required, nor even begun them in the 

 second, and that the more interesting, of the two cases to which 

 the method applies. I have employed the series contained in the 

 supplement before referred to, not however directly, for the pur- 

 pose of numerical calculation, but merely as stepping-stones en- 

 abling me to effect a certain analytical transformation in which the 

 use of series is got rid of. 



The method is not confined to the case of the highest possible 

 wave, but may also be used for lower waves, though unless the 

 wave is near the maximum it is better to have recourse to the 

 series. In any case of uniform propagation, we may readily reduce 

 the motion to a case of steady motion, and when that is done the 

 velocity of a particle at the surface will be the same as that of a 

 particle sliding along a smooth curve corresponding to the out- 

 line of the wave, and will accordingly be that due to the depth 

 below a fixed straight line, which for the sake of a name may be 

 called the datum line. In the case of the highest wave, since a 

 particle at the vertex of a wedge must be momentarily at rest, the 

 datum line will pass through the crest ; in other cases its height 

 above the wave must be assumed for trial as well as the outline of 

 the wave. 



The trial outline (and the trial datum line in the case of a 

 wave short of the highest) being known, the velocity at any point 

 of the surface is known, and therefore by an ordinary integration 

 or by a quadrature the velocity-potential at the surface is known. 

 Hence (f> being known in terms of x, x is known in terms of </>. 

 But the co-ordinates of points in the surface are given in terms of 

 cf> by equations (23), (24) of the supplement referred to on putting 

 yjr, the parameter of a stream line, = 0. These equations have 

 been simplified by choosing the units of space and time such as to 

 make a wave length, and also the change of </> in passing from one 

 wave to the next, each equal to 27r, and k is the value of — ^ at 

 the bottom. The equations then become 



x=-<j> + tA n (e nk +e-'" c ) sin ncf> (1). 



y = SA n (e" k -e- k ) cos n<f> (2). 



The negative sign of <j> in the first of these equations arises 



VOL. IV. PT. VI. 20 



