22 Prof. Powell's Abstract o/M, Cauchy*s 



Now, with regard to the attractive or repulsive forces, let 

 us suppose them to be proportional to the masses of the mole- 

 cules, and to some function of the distance f (r), which is posi- 

 tive in the case of attraction and negative in that of repulsion; 

 the mutual attraction of m tw will then be expressed by 



m m f (r) (3.) 



Then, (using the symbol S to represent the sum of a series 

 of corresponding terms referring to the molecules m, m\ m", 

 &c.) the resultant of the attractions or repulsions of the other 

 molecules upon m will have for its projections 



mS[m cos a f (r)], m S [m cos /3 f (r)], m S [tw cos y f (r)] (4.) 



and when the whole is in equilibrium, we shall consequently 

 have 



S [m cos « f (/•)] =0, S [w cos /3 f (r)] = 0, 



S [?« cos y f (r)] = (5.) 



Let us now suppose the equilibrium destroyed, and a mo- 

 tion to commence such, that the distance of the molecules m 

 m, shall vary in a ratio little different from unity. And at 

 the end of a time t, let the small displacements be ^, »), ^, re- 

 spectively parallel to the axes, and functions of a:,i/,z, t, whilst 

 the small displacement in the distance r is e. 



Thus we shall have a new set of values, corresponding to 

 each of the former expressions : we shall find the new coor- 

 dinates ofm, x-\-^ 7/ + ri 2 + ? 

 thoseof 7W, ^ + f+A(^ + f),^ + >j+AC3/ + >3), [2 + ?+ A(2+?) 

 whilst for r, we have r (1 +e) 

 and its projections A J7 + A f , A j/ +A>), Az+A? 

 and (substituting from (2.) the values of Ax, &c.) 



r^(l + ef = (r cos «+ A^T-\-{r cos /3+ Arif 



+ (rcosy+A?)^ (6.) 



Again, the cosines of the angles which the line joining 

 m m forms with the axes, will no longer have the values (2.), 

 but at the end of the time t will be represented by 



A^ + Ag _ cosa + ^- 



^(1+0 



A7/-hAYI 



r{i+e) 



Az+A K 



r{i-\-e) 1-f-f 



(7.) 



