SEQUEL TO THE GENERALIZATION. 379 



have been mentioned, lie and Lag-range treated the problems of the 

 small vibrations of fluids, both inelastic and elastic ; a subject which 

 leads, like the question of vibrating strings, to some subtle and ab- 

 struse considerations concerning the significations of the integrals of 



o o o 



partial differential equations. Laplace also took up the subject of 

 waves propagated along the surface of water ; and deduced a very 

 celebrated theory of the tides, in which he considered the ocean to be, 

 not in equilibrium, as preceding writers had supposed, but agitated by 

 a constant series of undulations, produced by the solar and lunar 

 forces. The difficulty of such an investigation may be judged of from 

 this, that Laplace, in order to carry it on, is obliged to assume a me- 

 chanical proposition, unproved, and only conjectured to be true-; 

 namely, 15 that, " in a system of bodies acted upon by forces which are 

 periodical, the state of the system is periodical like the forces." Even 

 with this assumption, various other arbitrary processes are requisite ; 

 and it appears still very doubtful whether Laplace's theory is either a 

 better mechanical solution of the problem, or a nearer approximation 

 to the laws of the phenomena, than that obtained by D. Bernoulli, 

 following the views of Newton. 



In most cases, the solutions of problems of hydrodynamics are not 

 satisfactorily confirmed by the results of observation. Poisson and 

 Cauchy have prosecuted the subject of waves, and have deduced very 

 curious conclusions by a very recondite and profound analysis. The 

 assumptions of the mathematician here do not represent the condi- 

 eions of nature ; the rules of theory, therefore, are not a good standard 

 to which we may refer the aberrations of particular cases ; and the 

 laws which we obtain from experiment are very imperfectly illustrated 

 by a priori calculation. The case of this department of knowledge, 

 Hydrodynamics, is very peculiar ; we have reached the highest point of 

 the science, the laws of extreme simplicity and generality from which 

 the phenomena flow ; we cannot doubt that the ultimate principles 

 which Ave have obtained are the true ones, and those which really 

 apply to the facts ; and yet we are far from being able to apply the 

 principles to explain or find out the facts. In order to do this, we 

 want, in addition to what we have, true and useful principles, inter- 

 mediate between the highest and the lowest ; between the extreme 

 and almost barren generality of the laws of motion, and the endless 

 varieties and inextricable complexity of fluid motions in special cases. 



i 5 Mec. &l. t. ii. p. 21S. 



