2 GWYTHER, Range of Progressive Waves in Deep Water. 



In order to obtain the expansions employed, I make 

 use of a solution of Lagrange's Equations of Fluid Motion 

 contained in a paper* read before this Society, of which 

 I reproduce a portion for the convenience of readers. 



The expressions which I find for the motion of the 

 fluid particles bring out prominently the general character 

 of the motion, namely that, as Stokes discovered, the 

 fluid particles have a progressive motion in the direction 

 of the wave propagation which diminishes rapidly as the 

 depth of the particles increases. This is necessary in 

 order that the fluid particles may reconcile themselves to 

 the conditions of the permanent progression, and, failing 

 this, the propagation cannot be continued. Since this 

 progressive motion increases with the height of the waves, 

 it seems probable that this circumstance plays a consider- 

 able part in practically determining the limit of their 

 height. 



The subject of the shape of the wave profile for waves 

 of different heights is of greater interest. In this section 

 of the paper it is shewn that if the co-ordinates of the 

 wave surface are expanded in a series of trigonometrical 

 terms, the coefficients of the successive terms diminish 

 very rapidly, the fall from the first to the second 

 coefficient being very remarkable. The results also 

 shew that the successive terms in the several coefficients do 

 not exhibit any marked tendency to converge. Hence, 

 although it is not desirable to take in many of the trigo- 

 nometrical terms, it may be desirable to have the co- 

 efficients of the terms retained worked out more fully- 

 than is here done. 



The last, and perhaps the more interesting, part of 

 the paper deals with the mathematical limit of the height 

 of the wave and the shape to which the profile in this 



* Manchester Memoirs, Vol. xliv. (1900), No. 10. 



