REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 143 



general solution from this integi'al, he makes a particular sup- 

 position, which is equivalent to considering the fluid to be de- 

 ranged from its state of equilibrium by causing the surface in 

 its whole extent to take the form of a trochoid, i. e. a serpentine 

 curve, of which the vertical ordinate varies as the cosine of the 

 horizontal abscissa. The solution in question is of so limited 

 a nature, that we may dispense with stating the results arrived 

 at. 



In the volume of the Memoirs of the Academy of Berlin for 

 the year 1786, Lagrange has given* a very simple way of 

 proving, in the Newtonian method of reasoning, that the ve- 

 locity of propagation of waves along a canal of small and con- 

 stant depth and uniform width, is that acquired by a heavy 

 body falling through half the depth. In the Mecanique Ana' 

 li/tiquef the same result is obtained analytically. The princi- 

 pal feature of the analysis in this solution is, that the linear 

 partial differential equation of the second order and of four va- 

 riables, to which the reasoning conducts, is integrated approxi- 

 mately in a series. Lagrange is of opinion, that on account of 

 the tenacity and mutual adherence of the parts of the fluid, the 

 motion extends only to a small distance vertically below the 

 surface agitated by the waves, of whatever depth the fluid may 

 be ; and that his solution will consequently apply to a mass of 

 fluid of any depth, and will serve to determine, from the ob- 

 served velocity of propagation, the distance to which the motion 

 extends downwards. 



The problem of waves was proposed by the French Institute 

 for the prize subject of 1816. M. Poisson, whose labours are 

 preeminent in every important question of Hydrodynamics, had 

 already given this his attention. His essay, which was the first 

 deposited in the bureau of the Institute, was read Oct. 2, 1815, 

 just at the expiration of the period allowed for competition. It 

 forms the first part of the memoir " On the Theory of Waves," 

 published in the volume of the Academy for the year 1816, and 

 contains the general formulas required for the complete solution 

 of the problem, and the theory, derived from these formulae, of 

 waves propagated with a nniformly accelerated motion. In the 

 month of December following, an additional paper was read by 

 M. Poisson on the same subject, which forms the second part 

 of the memoir just mentioned, and contains the theory of waves 

 propagated with a constant velocity. These are much more 

 sensible than the waves propagated with an accelerated motion, 

 and are in fact those which are commonly seen to spread in 



• p. 192. \ Part II. sect. xi. ait. 3G. 



