362 Prof. Stokes, On the highest wave [May 14, 



able height have been investigated by M. Boussinesq 1 and Lord 

 Rayleigh 2 . 



The evidence of the existence of a type of oscillatory irrota- 

 tional waves which are uniformly propagated is derived from the 

 nature of the process of approximation, which is one of a 

 systematic character that can certainly be carried on to as many 

 orders as we please, all the conditions of the problem being satis- 

 fied to the order to which we step by step advance. If therefore 

 the series that we are working with be convergent, there can be no 

 question of the possible existence of uniformly propagated waves. 

 But for a given depth and given wave length there remains in 

 our series a disposable constant on which the height depends, and 

 on the value of which the degree of convergency depends. By 

 taking this constant small enough the series will be convergent ; 

 though what the limit may be that separates convergency from 

 divergency, the process of expansion does not show. 



It seems to me pretty certain that the series will remain con- 

 vergent until a singular point appears at the boundary of the 

 fluid. Some years ago I was led by simple considerations to the 

 conclusion that the occurrence of a singular point in the profile at 

 which two branches meet at a finite angle (or as it might con- 

 ceivably have been touch, forming a cusp) entailed as a con- 

 sequence the existence of two tangents inclined at angles of 

 + 30° to the horizon ; so that the ridges of the waves came to 

 wedges of 120°. In a supplement to my former paper 3 lately 

 published I have conducted the approximation in a different 

 manner, which is more convenient for proceeding to a high order. 

 In this latter method the coordinates x, y are expressed in terms of 

 the velocity potential (}) and steam-line function ty, instead of cf> 

 and yjr in terms of x and y. The approximation is carried to the 

 fifth order for deep water, and to the third when the depth is 

 finite. Still even in this method the labour of the approximation 

 rapidly increases with the order, so that the result of working out 

 a great number of terms would not repay the labour ; and expan- 

 sion by series is hardly applicable to the determination of the cir- 

 cumstances of the highest possible wave. When a series whose 

 general term contains a power or other function of some parameter 

 a is convergent when a lies below a certain critical value, and 

 divergent when it lies above, it may be convergent when a has the 

 critical value, but if so its convergence is very slow. If we allow 

 that the highest possible wave comes to a ridge of 120°, that, com- 

 bined with our knowledge of the form of waves of very consider- 

 able height, would enable us to draw very approximately the 



1 Comptes Eendus, Tom. lxxii. (1871) p. 755. 



2 Philosophical Magazine, Vol. i. (1876), p. 257. 



3 Mathematical and Physical Papers, Vol. r. p. 314. 



