26 



which breaks upon our western coasts as a result of storms out in the Atlantic, 

 or the grand rollers which are occasionally observed at St. Helena and Ascension 

 Island. The motion in these cases having been produced from rest, by forces 

 applied to the surface, there is no molecular rotation, and therefore the investi- 

 gation of the present paper strictly applies. Moreover, if we conceive the waves 

 gradually produced by suitable forces applied to the surface, * * * the 

 investigation applies to the waves (secular change apart) at any period of their 

 growth, and not merely when they have attained one particular height. 



There can be no question, it seems to me, that this is the class of oscillatory 

 waves which on merely physical grounds we should naturally select for investi- 

 gation. The interest of the solution first given by Gerstner, and it is of great 

 interest, arises not from any physical preeminence of the class of waves to which 

 it relates, but from the imperfection of our analysis, which renders it important 

 to discuss a case in which all the circumstances of the motion can be simply ex- 

 pressed in mathematical terms without any approximation. And though this 

 motion is not exactly that which on purely physical grounds we should prefer to 

 investigate, namely, that in which the molecular rotation is nil, yet unless the 

 height of the waves be extravagant, it agrees so nearly with it that for many pur- 

 poses the simpler expressions of Rankine may be used without material error, 

 even when we are investigating wave motion of the irrotational kind. 



Stokes' criticism appears to be substantiated by the experimentally 

 determined fact that waves induce a mass transport of the magnitude 

 predicted by his theory. 



It is interesting to note that the work of Gerstner and Rankine 

 dealt only with waves in deep water. Gaillard's summary of the 

 trochoidal theory is extended to shallow-water waves, apparently by 

 grafting on the trochoidal surface form the conclusions of Stokes re- 

 garding the orbital and wave velocities. The antecedents of Gail- 

 lard's presentations are not clear from his text but he has been so 

 widely quoted that, regardless of historical background, Gaillard's 

 (16) book today represents the trochoidal theory however he may 

 have arrived at his equations. For ease of reference, all of the im- 

 portant equations of Gaillard will be quoted here with repetitions of 

 previous equations indicated by the equation numbers. 



Deep-water waves. — In terms of the method of generating the curve, 

 the trochoidal surface is described by the equations, 



x=Rd—r sin d (31a) 



y=R—r cos 9 (316) 



Here R is the radius of the rolling circle (L=2tR) and r is the radius 

 of the tracing circle (h=2r). The positions of crest and trough rela- 

 tive to still-water level are: 



Height of crest=|+0.7854^ (32a) 



Depth of trough=|- 0.7854^ (326) 



