18 



of the fluid are then considered, and the general equations applied 

 to a few simple cases. 



On considering these equations the author was led to observe, 

 that both Lagrange's and Poisson's proofs of the theorem that udx 

 + vdy + wdz is always an exact differential when it is so at any 

 instant (the pressure being supposed equal in all directions), would 

 still apply, whereas the theorem is manifestly untrue when the tan- 

 gential force is taken into account. This led him to perceive that 

 one objection to these proofs is of essential importance. He has 

 given a new proof of the theorem, which however was not necessary 

 to establish it, as it has been proved by M. Cauchy in a manner 

 perfectly satisfactory. 



The methods employed in this paper in the case of fluids apply 

 with equal facility to the determination of the equations of equili- 

 brium and motion of homogeneous, uncrystallized, elastic solids, the 

 only difference being that we have to deal with relative velocities in 

 the former case, and with relative displacements in the latter. The 

 only assumption which it is necessary to make, is that the pressures 

 are linear functions of the displacements, or rather relative displace- 

 ments, the displacements being throughout supposed extremely small. 

 The equations thus arrived at contain two arbitrary constants, and 

 agree with those obtained in a different manner by M. Cauchy. If 

 we suppose a certain relation to hold good between these constants, 

 the equations reduce themselves to Poisson's, which contain but one 

 arbitrary constant. 



The equations of fluid motion which would have been arrived at 

 by the method of this paper if the two constants ?, & had been re- 

 tained, have been already obtained by Poisson in a very different 

 manner. The author has shown, that according to Poisson's own 

 principles, a relation may be obtained between his two constants, 

 which reduces his equations to those finally adopted in this paper. 



There is one hypothesis made by Poisson in his theory of elastic 

 solids, by virtue of which his equations contain but one arbitrary 

 constant, which the author has pointed out reasons for regarding as 

 improbable. He has also shown that there is ground to believe that 

 the cubical compressibility of solids, as deduced by means of Pois- 

 son's theory from their extensibility when formed into rods or wires, 

 is much too great, a conclusion which he afterwards found had been 

 previously established by the experiments of Prof. Oersted. 



The equations of motion of elastic solids with two arbitrary con- 

 stants, are the same as those which have been obtained by different 

 authors as the equations of motion of the luminiferous eether in va- 

 cuum. In the concluding part of his paper the author has endea- 

 voured to show that it is probable, or at least quite conceivable, that 

 the same equations should apply to the motion of a solid, and to 

 those very small motions of a fluid, such as the aether, which accord- 

 ing to the undulatory theory constitute light. 



