740 Sir J. J. Thomson on the Application of the 



of the electrons are represented by the equations 



no =i pd, y = qd, z = rd, 



where p, q, r are all even and d is the shortest' distance 

 between an atom and an electron ; in the second class they 

 are represented by the equations 



x = pd, y = qd, z — rd, 



where two of the three integers are odd and^the third even. 

 The atoms are represented by 



x = pd, y = qd, z = rd, 



where p, q, r must either all be odd or two even and one 

 odd. 



We shall suppose that the displacements of the atoms 

 may be neglected. We shall denote a displacement parallel 

 to z of an electron of the first type by p, those of the second 

 type by p'. If we wish merely to investigate the stability, 

 it will be sufficient to confine our attention to the case which 

 is most likely to be unstable. A little consideration will show 

 that this is the one where the displacement of any electron is 

 equal and- opposite to that of its nearest neighbours in the 

 plane at right angles to the direction of displacement. This 

 will be the case if all the p displacements are the same and 

 if the p displacements are equal and opposite to the p ones 

 in the plane of xy and in all parallel planes whose distance 

 from xy is an even multiple of d, while in those in which the 

 distance is an odd multiple of d the p and p' are equal both 

 in sign and magnitude. 



In this case the force tending to increase p due to the 

 repulsion of the other electrons is equal to 



2pe 2 ^ / 1 3r 2 \ 



d' 6 \ [p 2 + q 2 + r 2 )* 2 (p 2 -rq 2 + r 2 f>V' 



where p and q must both be odd and r even. I find by 



a combination of arithmetic and integration that the quantity 



inside the bracket is equal to 1*04, so that the disturbing 



i , • 2'Otfpe 2 



force due to the electrons is — -^ — . 



Consider, now, the restoring effect due to the attraction of 

 the positive atoms. Owing to the symmetry of their distri- 

 bution round the electron, no term arises from the part of 

 the force which varies inversely as the square of the distance; 

 the whole effect is due to the part varying inversely as the 

 cube of the distance, and if we limit, as before, the effect of 



