428 Mr. J. H. Jeans on the 



of an atom which consists of a number of ions which is very 

 great but is not infinite. 



The ideal atom is to consist of a continuous distribution 

 of electricity of density p. The corresponding density of 

 matter is to be cr, so that it will ultimately be necessary to 

 put cr/p= +m/e, where mje is the ratio of the mass to the 

 charge, in the case of the negative ions of the cathode rays. 

 For the present, however, p and a will be regarded as capable 

 of independent variation. To correspond with the condition 

 of spherical symmetry for the actual atom, it will be assumed 

 that the distribution is arranged in spherical symmetry about 

 some centre, so that both p and a will be functions of the 

 single coordinate r, the distance from the centre. 



When the atom is at rest, every element is supposed to 

 act on every other element with a force which acts along 

 the line joining the two elements. The mutual potential of 

 two elements at distance r and of volumes dv dv' will be sup- 

 posed to be 



WxAf) + ™ r X2{r)}dv dv, 

 and for large values of r this must reduce to pp'r~ 1 dvdv'. 



When the atom is executing a small vibration, the forces 

 may be supposed to be the same as if the atom were at rest 

 in the displaced configuration. For in the case of a vibration 

 of infinitesimal amplitude, the electromagnetic forces will 

 vanish in comparison with the electrostatic, and it is easily 

 verified that the velocity of propagation may, without 

 appreciable error, be treated as infinite *. 



§ 13. Let the whole atom undergo a small continuous dis- 

 placement, so that the element of which the spherical polar 

 coordinates before displacement were r, 0, (f> is moved to 

 r + u, 6 + v, <f> + w. In rectangular coordinates, which it will 

 be convenient to use in conjunction with polars, let the 

 element initially at x, y, z be displaced to x + g, y + 7), z + f. 



Consider, in the first place, the field of force arising from 

 the term Xi( r ) ^ n the potential-function. The displacement 

 of the element p' da' dy' dz' is equivalent to the creation of a 

 doublet of moment 



(f p' dx ! dy' dz\ r/ p' doc' dy 1 dz', £' p' dx } dy 1 dz') . 

 Hence the potential per unit charge, upon the element 

 originally at x, y, z, but now at x + %, y + v> z+Z, which 

 arises from the displacement of the foregoing element, will be 



pW^'{(?'-?)^ + W-v)^, + (C-B ^>}xi(R) 



where R?= (x-x') 2 + (y-y' ) 2 + (z-z 1 ) 2 . 



* Larmor, Carnb. Phil. Tran*. xviii. p. 391. 



