ON WAVES. 331 



ing the general equations for the motion of incompi-essible fluids in the 'Me- 

 canique Analytique,' part 2. sect, ix., Lagrange says, " Voila les formules 

 les plus generales et les plus simples pour la determination rigoureuse du 

 mouvemt-nt des fluides. La diffieulte ne consiste plus que dans leur integra- 

 tion;" and then he adds elsewhere, " malheureusement elles sont si rebelles, 

 qu'on n'a pu jusqu'a present en venir a bout que dans des cas tres-limites." 

 Indeed, ever since the publication of Euler's general formula for the motion 

 of fluids in the Memoirs of the Academy of Sciences of Berlin, 1755, tha 

 whole phaenomena of fluids in all conditions may be considered as having 

 been represented. But the phaenomena have remained there till now, locked 

 up without any one to open, and amongst the rest I pi-esume the wave of the 

 first order. 



There is one point, however, in which the analysis of M. Lagrange has 

 appeared to make an approach to the representation of one of the phaenomena 

 peculiar to the wave of translation. In section xii. of part 2. of the ' Me- 

 canique Analytique,' he investigates the propagation of vibrations in elastic 

 fluids (like those of sound through the atmosphere), and obtains an equation 



d^<t>_ 





from which he afterwards deduces the well-known law that sound is propa- 

 gated with a velocity (nearly) equal to that which is due to gravity, acting 

 freely through a height equal to half the depth of the atmosphere (supposed 

 homogeneous and of uniform density). And again, elsewhere he finds for the 

 propagation of wave motion in a liquid in a channel with a level bottom, and 

 a depth a, the equation 



dt- \d^" dx-J 



and from the similarity of this to the former equation, he argues as follows : 

 " Ainsi comme la vitesse de la propagation du son se trouve 6gale a celle qu'un 

 corps grave acquerrait en tombant de la moitie de la hauteur de I'atmosphere 

 supposee homogene, la vitesse de la propagation des ondes sera la meme que 

 celle qu'un corps grave acquerrait en descendant d'une hauteur egale a la 

 moitie de la profondeur de I'eau dans le canal." 



Had this result been of the same general nature with the original equations 

 from which it is deduced, we should have been able to assign to the analysis 

 of M. Lagrange the honour of having predicted in 1815 the wave of the first 

 order, never distinctly recognised by observation till 1834. Unhappily the 

 nature of his investigation precludes us from doing so, and he goes on himself 

 to admit that this conclusion will only apply to such waves as are infinitely 

 small, and agitate the water to a very small depth below the surface. " On 

 pourra toujours employer la theorie precedente, si on suppose que dans la 

 formation des ondes I'eau n'est 6branlee et remuee qu'a une profondeur tres- 

 petite." Tlie wave of the first order bears as its characteristics, the observed 

 phaenomena, that the agitation does extend below the surface to the very 

 bottom of the channel, where it is quite as great as at the surface, and that 

 its oscillations are large. The essential conditions of Lagrange's analysis be- 

 ing that the oscillation is minute, and that the agitation of the fluid is con- 

 fined to the surface, we are precluded from the application of his formula to 

 the wave of the first order. 



I have been led to speak thus fully of M. Lagrange's solution, because his 

 result is the only one that off'ers a tolerable approximation to the represen- 

 tation of the velocity of the wave of the first order. I do not find in the re- 



