84 REPOHT— 1881. 



to develop a solTiewliat similar theory for a fluid of finite depth. In tlie 

 trocboidal waves in deep fluids all the jDarticles describe circles with 

 uniform velocities. Hagen starts by assuming the path to be an ellipse 

 distorted so that the higher and lower halves are raised from the sym- 

 metrical position, i.e. the paths are given by 



a; = a sin f, y = ft cos f + y cos^ f 



where ^ is a function of the time and cij3,y are constants for each 

 particle. He also assumes that particles in a vertical line always remain 

 so. The condition of constant pressure at the surface as expressed by 

 him is only approximately satisfied. St. Venant' has given for stationary 

 waves a theory similar to that of Gerstner's for progressive waves, and 

 has also attempted to develop a theory for progressive and stationary 

 waves when the fluid is of a finite depth, but unfortunately his theory 

 does not satisfy the equation of continuity. 



The limiting form of trochoidal waves being the cuspidal, it is clear 

 that permanent waves could exist whose crests would be infinitely fine, a 

 result contradicted by experience if we suppose the theory actually to 

 apply to waves as ordinarily seen, that is in irrotational motion. This 

 alone would show that they cannot be taken as the type of naturally 

 produced waves. In another wave-form considered by Rankine,^ 



•4/ = 2mj/X - e.vj} (- 2-7/ /A) cos 27rxl\ 



the steepest form cuts itself at 90° ; but this is a forced wave, that is the 

 pressure along the surface is not constant. The velocity of propagation 

 here also is \/((7\/27r). In a supplement to this paper^ he attempts to 

 prove that all waves in which molecular rotation is null must begin to 

 break when the two slopes of the crest meet at a right angle. But this 

 has been criticised by Stokes' who has very neatly proved that if such a 

 sharp angle is possible it must be one of 120° ; and this is borne out by 

 the fact that the analytical series for the permanent type of irrotational 

 periodic waves seems to become divergent as the wave approaches the 

 form with a crest of 120°. 



The question of permanent types of waves of longitudinal disturbance 

 through the medium, properly belongs to the domain of the theory of 

 sound. It has been investigated by Stokes,* Earnshaw,*" Riemann,^ and 

 Rankine.^ The essential difference between this case and the preceding 

 is that the medium must fulfil the condition (t^djjjdp = const. Earnshaw 

 regarded this as unrealisable ; but Rankine has shown that with a given 

 law of conduction of heat, there are types of waves which can be pro- 



• ' Sur la houle et le clapotis,' Compt. Bend., Ixxiii., p. .521, 589. 



^ ' Summary of the properties of certain stream-lines,' ridl. Mag. (4), xxviii. 

 p. 282. 



^ Phil. Mag. (4), xxix., p. 25. 



* Math, and Phys. Papers. App. B., ' Considerations relative to tlie gi'eatest 

 lieight of oscillatory irrotational waves wliich can be propagated without change Of 

 form,' p. 225. 



= ' On a difficulty in the theory of sound,' PMl. Mag. (.^), xxxiii. p. .^49. 



^ ' On the mathematical theory of sound,' Trans, hoy. Soc. (1860), p. 1.^.3. 



^ 'Ueber die Fortpflanzung ebener Luftwellen,' Gott., Ahhand. Math. Class., viii. 

 p. 43. 



" ' On the thermodynamic theory of waves of finite longitudinal disturbance,' 

 Trans. Roy. Soc, 1870. Also Reprint, p. 530, 



