1883.] Prof. Stokes, On the highest wave of 'uniform propagation. 361 



The following communications were made to the Society : — 



(1) On the highest ivave of uniform propagation, Preliminary 

 notice. By G. G. Stokes, M.A., F.R.S., Lucasian Professor of 

 Mathematics in the University of Cambridge. 



There is one particular case of possible wave motion, appli- 

 cable to a fluid of practically infinite depth, in which all the 

 circumstances of the motion admit of being expressed mathe- 

 matically in finite terms, the necessary equations being satisfied 

 exactly, and not approximately only ; while the general expressions 

 contain an arbitrary constant permitting of making the amplitude 

 any whatsoever up to the extreme limit of cycloidal waves, coming 

 to cusps at the crests. This possible solution of the equations was 

 given first by Gerstner, near the beginning of the present century. 

 The motion however to which it relates is not of the irrotational 

 class, and could not therefore be excited in a fluid previously at 

 rest by forces applied to the surface ; nor could it be propagated 

 into still water from a disturbance at first at a distance. In fact, 

 the conditions requisite for its existence are of a highly artificial 

 character ; so that the chief interest of the solution is one arising 

 from the imperfection of our mathematics, which makes it desirable 

 to discuss a case of possible motion, however artificial the condi- 

 tions may be, in which everything relating to the motion can be 

 pretty simply expressed in finite terms. 



There can be no question however that it is the irrotational 

 class of possible wave motions which possesses the greatest, almost 

 the only, intrinsic interest ; since it is this kind alone which can 

 be excited in a fluid previously at rest by means of forces applied 

 to the surface, such for example as the unequal pressure of the 

 wind on the surface, or propagated into previously still water from 

 a distance. 



In a paper read before the Society in 1847, and published in 

 the transactions, I have investigated the motion of oscillatory 

 waves in which the motion is not very small by the method of 

 successive substitutions, proceeding to the second order in the case 

 of an arbitrary depth, and to the third order in the simpler case in 

 which the depth is infinite. In the latter case the terms of the third 

 order were found to be very small even in the case of waves of 

 very considerable magnitude. The series converge less rapidly 

 when the depth is finite ; and when the length is very great 

 compared with the depth of the fluid the convergence becomes so 

 slow that the method practically fails, and is not therefore appli- 

 cable to solitary waves. 



The circumstances of the motion of solitary waves of consider- 



