REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 147 



Memoirs of the Academy concludes with a bi'ief account of the 

 history and principle of this way of expressing the complete 

 integral by a series of particular integrals, and introducing the 

 arbitrary function. But 1 would chiefly recommend the peru- 

 sal of the remarks at the end of a memoir by this author " On 

 the Integration of some linear partial Differential Equations ; 

 and particularly the general Equation of the Motion of Elastic 

 Fluids." To the memoir itself I beg to refer, by the way, as 

 presenting a demonstration of the constancy of the velocity of 

 propagation from an irregular disturbance in an elastic fluid, 

 more simple and direct than that in the Journal de VEcole Po- 

 lytechnique. It contains also a general integral of the linear 

 partial differential equation of three terms, which occurs in the 

 problem of waves for the case in which the three dimensions of 

 the fluid are taken account of; but the author does not consider 

 this integral of much utility, because of the impossible quantities 

 involved in it, and rather recommends the method of express- 

 ing the principal variable by infinite series of exponentials. In 

 fact, in the *' Theory of Waves " this case is treated in a manner 

 exactly analogous to that in which abstraction is made of one 

 dimension of the fluid. 



It may be useful to state some of the principal results ob- 

 tained by theory respecting the nature of waves, to give an idea 

 of what the independent power of analysis has been able to ef- 

 fect. 



With respect, first, to the canal of uniform width, the law of 

 the velocity of pi'opagation found by Lagrange is confirmed by 

 M. Poisson's theory when the depth is small, but not other- 

 wise. 



When the canal is of unlimited depth, the following are the 

 chief results : 



(1.) An impulse given to any point of the surface affects in- 

 stantaneously the whole extent of the fluid mass. The theory 

 determines the magnitude and direction of the initial velocity of 

 each particle resulting from a given impulse. 



(2.) " The summit of each wave moves with a uniformly acce- 

 lerated motion." 



This must be understood to refer to a series of very small 

 waves, called by M. Poisson dents, which perform their move- 

 ments as it were on the surface of the larger waves, which he 

 calls " les ondes denteUes" Each wave of the series is found 

 to have its proper velocity, independent of the primitive im- 

 pulse. Waves of this kind have been actually observed : they 

 are small from the first, and quickly disappear. 



(3.) At considerable distances from the place of disturbance,^ 

 l2 



