Prof. Thomson, On the vibrations of atoms, etc. 39 



On the vibrations of atoms containing 4, 5, 6, 7, and 8 corpuscles 

 and on the effect of a magnetic field on such vibrations. By 

 J. J. Thomson, M.A., F.R.S., Cavendish Professor of Experimental 

 Physics, Cambridge. 



[Received 27 December 1904.] 



The paper contains an investigation of the time of vibration of 

 model atoms of some complexity, it was undertaken in the hope 

 that the properties of these models might throw some light on the 

 effect of a magnetic field on the lines in the spectra of luminous 

 bodies. 



Case of Four Corpuscles. 



We suppose that we have four equal corpuscles with equal 

 charges of negative electricity placed within a sphere of uniform 

 positive electrification; when in equilibrium the corpuscles will be 

 situated at the corners of a regular tetrahedron with its centre at 

 the centre of the sphere. We can easily show that if b is the 

 radius of the sphere of positive electrification, and a the distance 

 of any one of the corpuscles from the centre of the sphere, then 

 when the attraction of the positive sphere on the corpuscle is equal 

 and opposite to the repulsion exerted by the other three corpuscles 



a 3 = 3V3 

 P"16V2' 



the positive charge in the sphere being equal in magnitude to the 

 sum of the negative charges on the four corpuscles. This result 

 may be expressed by saying that when in equilibrium the distance 

 between two corpuscles is equal to b. 



Let us call the equilibrium positions of the four particles 1, 2, 

 3, 4 and let us take as our axes of coordinates the lines joining 

 the middle points of opposite sides of the tetrahedron. The line 

 joining the middle points of (12), (34); (13), (24); (14), (23) being 

 taken as the axes of x, y, z respectively ; the coordinates of the 

 points 1, 2, 3, 4 are respectively (a, a, a), (a, — a, — a), (— a, a, — a), 

 (— a, — a, a), where 3a 2 = a 2 . 



If (fj r , v r , £.) are the displacements of the rth particle from its 

 position of equilibrium, m the mass and e the charge on a corpuscle, 

 we can show that the vibrations of the system are given by the 

 equations : 



/-ix d % P 7 d 2 q . d 2 r , 



(1) m -£ = - kp > m di? = - kq > m d? = - kr > 



