PROFESSOR KELLAND ON THE THEORY OF WAVES. 4yy 



Newton, in the Second Book of his Principia, apphed his power of ab- 

 stracting the point adapted to computation from the mass around it, to the sohi- 

 tion of tlie problem of oscillation in a deep fluid, and his results are still valuable, 

 as his process is ingenious. 



Laplace afterwards applied a regular analysis to the problem,* but his so- 

 lution is of a very Umited nature. Lagrange, in his Mecanique Analytique,-\ 

 gives equations which contain the complete statement of the question, and all 

 that remains is the treatment of these equations. Lagrange himself :|: correctly 

 solves the problem for fluids of small depth, but his hypothesis is very far from 

 the truth as applied to the general problem. 



It was in this state of the question that the French Institute proposed the 

 subject as its prize essay for 1816. 



M. PoissoN, who, as he himself states, had for a long time been engaged on 

 this problem, sent first a memoir to the Institute in October 1815, and afterwards 

 a second in December. These memoirs contain an approximate solution of the 

 motion of waves in a canal. We commence first with M. Poisson's.^ 



This most important memoir starts with an approximation which (M. Challis 

 thinks) aU persons who have attempted the problem have used. The assmnp- 

 tion which gives rise to his equation (4), p. 81, also appears to riie most essen- 

 tially to destroy the generality of the results. As to the body of the memoir it- 

 self, it contains very little which belongs to'our present matter. It is principally 

 occupied in discussing the question of the effects which presently foUow a disturb- 

 ance in the fluid. The theoretical results of M. Poisson have been analyzed and 

 tested by M. Weber, with whose work I am altogether unacquainted. 



The prize mentioned above was adjudged to M. Cauchy, who was thought 

 to have solved the question more generally. As to what the problem was I am 

 not altogether certain. In M. Cacjchy's memoir, printed in 1827, || fifteen years 

 after it had obtained the prize, I find it stated thus : 



" Une masse fluide pesante, primitivement en repos, et d'un profondeur inde- 

 finie, a ^t^ mise en mouvement par Teffet d'une cause donn^e. On demande, au 

 bout d'un temps d^termin^, la forme de la surface exterieure du fluide, et la Vi- 

 tesse des mol^ciiles situ^es a cette m^me surface." 



I have little doubt this is the form in which the question was actually pro- 

 posed, and we find a sufficient cause for the detention of the two philosophers on 

 this part of the problem, to the exclusion, or nearly so, of others. M. Cauchy 

 has added notes to his memoir, partly to explain and partly to enlarge on the 



* Memoires de I'Academie des Sciences, 1776. 



+ Mecanique Analytique, 2d Partie, Sect. xi. % Ibid., Arts. 35-39. 



§ Memoires de I'Academie des Sciences, 1816. || Memoires des Savans Etrangers, tome i. 



