AND THE EQILIBRIUM AND MOTION OF ELASTIC SOLIDS. 313 



more remote, of which the average may be taken. The consequence of this supposition of course is 

 that the total action is normal to the base of the hemisphere, and sensibly the same for one 

 . molecule as for an adjacent one. 



The rest of the reasoning by which Poisson establishes the equations of motion and equililiriuuj 

 of elastic solids is purely mathematical, sufficient data having been already assumed. It might 

 appear that the reasoning in Art. ifi of his memoir, by which the expression for A'^ is simplified, 

 required the fresh hypothesis of a symmetrical arrangement of the molecules; but it really does not, 

 being admissible according to the principle of averages. Taking for the natural state of the body 

 that in which the pressure is zero, the equations at which Poisson arrives contain only one 

 unknown constant k, whereas the equations of Sect. iii. of this paper contain two, A or m A and B. 

 This difference depends on the assumption made by Poisson that the irregular part of the force 

 exerted by a hemisphere of the medium on a molecule in the centre of its base may be neglected in 

 comparison with the whole force. As a result of this hypothesis, Poisson finds that the change 

 in direction, and the proportionate change in length, of a line joining two molecules are continuous 

 functions of the co-ordinates of one of the molecules and the angles which determine the direction of 

 the line ; whereas in Sect. III., if we adopt the hypothesis of ultimate molecules at all, it is 

 allowable to suppose that these quantities vary irregularly in passing from one pair of molecules to 

 an adjacent pair. Of course the equations of Sect. iii. ought to reduce themselves to Poisson's 

 equations for a particular relation between A and B. Neglecting the heat developed by compression, 

 as Poisson has done, and therefore putting m = 1, this relation is ^ = 5fl. 



18. Poisson's theory of fluid motion is as follows. The time t is supposed to be divided 

 into a number n of equal parts, each equal to t. In the first of these the fluid is supposed to 

 be displaced as an elastic solid would be, according to Poisson's previous theory, and therefore 

 the pressures are given by the same equations. If the causes producing the displacement were 

 now to cease, the fluid would re-arrange itself, so that the average arrangement about each point 

 should be the same in all directions after a very short time. During this time, the pressures 

 would have altered, in an unknown manner, from those corresponding to a displaced solid to a 



normal pressure equal to p -t- f—T, the pressures during the alteration involving an unknown 



function of the time elapsed since the end of the interval t. Another displacement and another 

 re-arrangement may now be supposed to take place, and so on. But since these very small 

 relative motions will take place independently of each other, we may suppose each displacement to 

 begin at the expiration of the time during which the preceding one is supposed to remain, and we 

 may suppose each re-arrangement to be going on during the succeeding displacements. Supposing 

 now n to become infinite, we pass to the case in which the fluid is supposed to be continually 

 beginning to be displaced as a solid would, and continually re-arranging itself so as to make the 

 average arrangement about each point the same in all directions. 



Poisson's equations (9), page 1.52, which are applicable to the motion of a liquid, or of an 

 elastic fluid in which the change of density is small, agree with equations (12) of this paper. For 

 the quantity \^t is the pressure p which would exist at any instant if the motion were then to 



cease, and the increment, t or ^^t, of this quantity in the very small time t will depend 



1 1 ■ ^"xt Dp ,. , , . ,, 11,,. d.^t 



only on the increment, — ^^t or — — t, of the density \t or p. Consequently the value ot — ■ — t 

 dt Dt J M I H J ^f 



dyt 

 will be the same as if the density of the particle considered passed from j^< to ;^< + -\^ t in the 



time T by a uniform motion of dilatation. I suppose that according to Poisson's views such a 

 motion would not require a re-arrangcment of the molecules, since the pressure remains, equjil 

 Vol.. VIII Paui III. Ss 



