44 



Prof. Th 



>.omson, On the vibrations 



of 



atoms 



where Xj 



, \ 2 are the 



roots of the 

 V = x(|- 



equation 

 8^8/ ' 







or approximately 



x, 



_ 9 







The< 



corresponding periods are 











*Vs- 



2 Vi' 







where 





A^j == i o A/^ 



& 2 = -038 k. 







Effect of a magnetic field on the periods. The effects on the 

 vibrations corresponding to x, y, z are normal, that on 



Pi + q2 + r z , pi-q 2 , Pi-r 3 



is zero, while the effects on the coordinates 



r + \ (p 2 + qi), r + \ 2 (p 2 + qj, 



are respectively about ^ and ^ of the normal. 



Case of Six Corpuscles. 



The six particles when in equilibrium will be at the corners of 

 a regular octahedron. Denoting the positions of the corpuscles 

 when in equilibrium by the symbols 1, 2, 3, 4, 5, 6, let us take the 

 line joining 1 and 6 as the axis of x, that joining 2 and 5 as the 

 axis of y and that joining 3 and 4 as the axis of z. If a is the 

 distance of any one of the corpuscles in its undisturbed position 

 from the centre of the sphere of positive electrification then the 

 coordinates of the points 1, 2, 3, 4, 5, 6 are as follows 1 (a, 0, 0), 

 2 (0, a, 0), 3 (0, 0, a), 4 (0, 0, - a), 5 (0, - a, 0), 6 (- a, 0, 0); let 

 £r, Vr> Kr be the displacement of the rth corpuscle from its position 

 of equilibrium, then, m being the mass and e the charge on a 

 corpuscle, the oscillations of the corpuscles will be given by the 

 following equations 



d 2 p , d 2 q , d 2 r 7 



m M = ~ kp > m M=- k( *> m d? = - kr > 



where p = & + f 2 4- f? f + & + & + &, 



q = Vi + V2 + Vs + Vi + Vs + Vsl 

 r=$i + r 2 + r 3 + £ 4 + £ 6 +£ 6 , 



