REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 145 



sion and sudden elevation of bodies of the forms of a parabo- 

 loid, a cylinder, a cone, and a solid, generated by the revolution 

 of a parabola about a tangent at its vertex. To bodies of the 

 last three forms, M. Poisson objects to extending the reasoning; 

 and in the " Note" above referred to, attempts to show that such 

 an extension leads to results inconsistent with the principle of 

 the coexistence of small vibrations. If we are not permitted to 

 receive the analysis of M. Cauchy in all the generality it lays 

 claim to, we must at least assent to the reasonableness of the 

 following conclusion it pretends to arrive at, viz. that "the 

 heights and velocities of the different waves produced by the 

 immersion of a cylindrical or prismatic body depend not only 

 on the width and height of the part immersed, but also on the 

 form of the surface which bounds this part." There is also 

 much appearance of probability in a remark made by the 

 same mathematician, that the number of the waves produced 

 may depend on the form of the immersed body and the depth 

 of immersion. 



We proceed to say a few words on the contents of M. Pois- 

 son's memoir. He commences by showing, as well by a priori 

 reasoning as by an appeal to facts, that Lagrange's solution 

 cannot be extended to fluid of any depth. In his own solution 

 he supposes the fluid to be of any uniform depth, but princi- 

 pally has regard to the case which most commonly occurs of a 

 very great depth : he neglects the square of the velocity of the 

 oscillating particles, as all have done who have attempted this 

 problem, and assumes, that a fluid particle which at any instant 

 is at the surface, remains there during the whole time of the 

 motion. This latter supposition seems necessary for the con- 

 dition of the continuity of the fluid. With regard to the neg- 

 lect of the square of the velocity, it does not seem that we can 

 tell to what extent it may affect the calculations so well as in 

 the case of the vibrations of elastic fluids, where the velocity of 

 the vibrating particle is neglected in comparison of a known and 

 constant velocity, that of propagation. M. Poisson treats first 

 the case in which the motion takes place in a canal of uniform 

 width, and, consequently, abstraction is made of one horizontal 

 dimension of the fluid ; and afterwards the case in which the 

 fluid is considered in its three dimensions. The former requires 

 for its solution the integration of the same differential equation 

 of two terms * as that occurring in Laplace's theory. No use 

 is made of the common integral of this equation, as, on account 

 of the impossible quantities it involves, it would be difficult 



* In M. Poissoii's works this equation is '-1? -f — = 0, 



1833. T. 



