382 Lord Uayleigh on Periodic Jrrotational 



50 per cent, from that given by Stokes' formula, viz. 

 fi=— \o£. It seems to me that too much was expected. 

 A series proceeding by powers of \ need not be very 

 convergent. One is reminded of a parallel instance in the 

 lunar theory where the motion of the moon's apse, calcu- 

 lated from the first approximation, is doubled at the next 

 step. Similarly here the next approximation largely in- 

 creases the numerical value of /3. When a smaller a is 

 chosen (yo), series developed on Stokes' plan give satis- 

 factory results, even though they may not converge so 

 rapidly as might be wished. 



The question of the convergency of these series is distinct 

 from that of the existence of permanent waves. Of course a 

 strict mathematical proof of their existence is a desideratum; 

 but I think that the reader who follows the results of the 

 calculations here put forward is likely to be convinced that 

 permanent waves of moderate height do exist. If this is so, 

 and if Stokes' series are convergent in the mathematical 

 sense for such heights, it appears very unlikely that the 

 case will be altered until the wave attains the greatest 

 admissible elevation, when, as Stokes showed, the crest conies 

 to an edge at an angle of 120°. 



It may be remarked that most of the authorities men- 

 tioned above express belief in the existence of permanent 

 waves, even though the water be not deep, provided of 

 course that the bottom be flat. A further question may be 

 raised as to whether it is necessary that gravity be constant 

 at different levels. In the paper first cited I showed that, 

 under a gravity inversely as the cube of the distance from 

 tiie bottom, very long waves are permanent. It may be that 

 under a wide range of laws of gravity permanent waves 

 exist. 



Following the method of my paper of 1911, we suppose 

 for brevity that the wave-length is 2ir, the velocity of pro- 

 pagation unity*, and we take as the expression for the stream- 

 function of the waves, reduced to rest, 



-^r=y— ae~ y cos x—/3e~ 2 v cos2ti—ye~ Sl/ cos Zoc 



— Se-^cosAx — ee-^cosSx, . . (1) 



in which # is measured horizontally and ?/ vertically downwards. 

 This expression evidently satisfies the differential equation 



f; * The extension to arbitrary wave-lengths and velocities may be 

 effected at any time by attention to dimensions. 



