48 Prof. Thomson, On the vibrations of atoms, etc. 



then 



4 + 1 



, , \/2 3 \/3 , 

 where ^ = - ^— A:, 



2 + »/2 + — — 

 , 1 2 



The arrangement of eight corpuscles at the corners of a cube 

 is however unstable, for, if 



p = X 1 + X 2 -(X 3 + X i ), 



q =Y 1 +Y 3 ~(Y 2 +Y i ), 



r = Z 1 + Z i -(Z 2 + Z 3 ), 



we find 



3 2 



and, since 2 — — , is positive, the cubical arrangement is 



unstable for a displacement when p — q has a finite value. We 

 can show also that the cubical arrangement is unstable, even 

 although a corpuscle is placed at the centre. 



The stable arrangement for eight particles is when we have 

 six particles at the corners of an octahedron and two others at 

 equal distances from the centre on one of the lines joining 

 opposite corners of the octahedron ; the octahedron will no longer 

 be regular, the axis along which the two extra particles are placed 

 being longer than the other two, approximately in the proportion 

 of 4 to 3, the distance of the inner particles from the centre is 

 about | of the distance of the furthest corner of the octahedron 

 from the centre. 



