REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 139 



density and velocity, at the same distance in every direction 

 from a fixed point, which is the centre of it. Laplace first dis- 

 pensed with this limitation in the case in which two dimensions 

 only of the fluid are taken account of*. The principal cha- 

 racter of his analysis is a new method of employing definite 

 integrals. Finally, M. Poisson solved the same problem for 

 three dimensions of the fluid f. This memoir deserves to be 

 particularly mentioned for the interesting matter it contains. 

 The object of the author is to demonstrate, in a more general 

 manner than had been before done, some circumstances of the 

 motions of elastic fluids which are independent of the particular 

 motions of the fluid particles, such as propagation and reflection. 

 The general problem of propagation just mentioned he solves 

 by developing the integral of the partial differential equation of 

 the second order in x, y, z, and t, applicable to this case, in a 

 series proceeding according to decreasing powers of the di- 

 stance from the centre of disturbance, as it cannot be obtained 

 in finite terms, and then transforming the series into a definite 

 integral, — a method which has of late been extensively em- 

 ployed. The crossing of waves simultaneously produced by 

 disturbances at several centres, is next considered, and this 

 leads to a general solution of the problem of reflection at a plane 

 surface. For the case in which the motions of the aerial parti- 

 cles are not supposed small, the velocity of propagation along 

 a line of air is shown to be the same as when they are small. 

 This result is an inference drawn from the arbitrary disconti- 

 nuity of the motion, on which it does not seem to depend. In 

 a paper before alluded to J, the same result is obtained without 

 reference to the principle of discontinuity. M. Poisson treats 

 also of propagation in a mass of air of variable density, such as 

 the earth's atmosphere. His analysis is competent to prove, in 

 accordance with experience, that the velocity of sound is the 

 same as in a mass of uniform density, and that its intensity at 

 any place depends, in addition to the distance from the point 

 of agitation, only on the density of the air where the disturbance 

 is made. So that a bell rung in the upper regions of the air 

 will not sound so loud as when rung by the same effort below, 

 but will sound equally loud at all equal distances from the place 

 where it is rung. 



In seeking for the general equations of the motion of fluids, 

 (first obtained by Euler,) a quantity § is met with which, if it be 



* Memoires de V Academic, An 1779. , 



t " Memoire sur la Theorie du Son," Journal de I'Ecole Polytechnique, 

 torn. vii. cah. xiv. 



J Transactions of the Philosophical Society of Cambridge, vol. iii. Part III. 

 § In M. Poisson's writings this quantity is udx + v dy + ir dz. 



