AS EXPLAINED BY THE HYPOTHESIS OF FINITE INTERVALS. 157 



varying according to any function of the distance. It would be a use 

 less generalization, in the present state of analysis, to proceed at once 

 to the solution of this problem without any further restrictions, for 

 even should we succeed in integrating the resulting equations, whether 

 by approximation or any other method, we should at length be obliged 

 to have recourse to particular hypotheses in order to interpret tur 

 results. 



1 propose then to make the following hypotheses: 



1. That the distance between the particles is sufficiently large com- 

 pared with their sphere of motion, to allow the square of the latter 

 quantity to be omitted compared with that of the former. 



2. That the disposition of the particles is a disposition of symmetry. 

 It may serve to fix our ideas, if we consider them symmetrically situ- 

 ated with respect to the three co-ordinate planes; as, for instance 

 arranged in the angular summits of cubes, whose edges are parallel to 

 the co-ordinate axes, and whose centre is the origin. This is, however 

 merely stated as something to guide us, since we must suppose in 

 whatever manner it can be accomplished, that the disposition is ' lier- 



fectly symmetrical. On these two hypotheses, which, virtually at least 

 are M. Cauchy's hypotheses, I shall now proceed to determine thJ 

 equations of motion. 



Let X, y, % be the co-ordinates of any particle P in its state of rest 

 the origin being taken at pleasure, and the axes any axes of symmetry! 

 X ^Ix, y ^hj, s: + 1% those of any other particle Q, which lies within 

 the sphere of sensible attraction to P; r the distance PQ; x + a, y ^ p, 

 z + 7 the co-ordinates of P after any time t from the beginning' of thJ 

 motion ; x + a + Sx + Sa, y + ^ + Sy + Sfi, z + y + S^ + ^y thote of Q 

 at the same time ; r + p' the corresponding value of the distance PQ 

 Let the accelerating force of Q on P at the distance PQ be represented 

 by the function {r + p') .cj>(r + p'). Resolving this attraction parallel to 

 the axis of x, it gives (r + p) . (Sx + Sa), whence we obtain 



