254 Mb KELLAND, ON THE MOTION OF 



whence the above expression becomes 



n 2 N TT 1 . . irXt 



Tj)2-^-^"^^^ 



^•■/-U7TT- (. + i).(.-- '-~--^'" 



the coefficient in which contains — — r . 



e 



/ 2 \ 1 



This owes its value to the factor m - A — -. 



\ n - 11 6"-' 



We could not, consistently witii the restriction before imposed on 

 this quantity that it shall be small, suppose u = 1, as that hypo- 

 thesis would make it very large, and, what is worse, would affect it 

 with the negative sign. 



If w be a considerable quantity, the factor of -^, is not large, 



but — r is itself considerable. 



g« - 1 



If M = 2 the expression is reduced to zero, and the velocity will 

 be found by extending the expansions in finding the finite integrals, 

 and retaining the smaller terms, and will consequently be very small. 



The condition which we required, therefoie, that the velocity should 

 be very small is satisfied, and, apparently, only satisfied by the hypo- 

 thesis of the force varying inversely as the square of the distance. 



21. It may not then be uninteresting to examine this law of the 

 force a little more closely, although it appears by no means probable 

 that the statical condition of the pressure varying as the density, can 

 be reduced to this law, or indeed to any other, properly so called. 



The investigation which follows, of the statical condition of a system 

 of particles exerting repulsive energies of this nature, will sufficiently 

 prove this, if proof be necessaiy. 



