- -. ; (0i + (f>2 + fa) = - K (0 X + </> 2 + ^ 3 ), 



d 2 

 ■-„ (0i-4> 2 ) = - h (0i - </> 2 ), to -rr 2 (0i- ^3) = - & 5 (#i ~ -f s)- 



46 Pro/! Thomson, On, the vibrations of atoms 



Thus <£ 3 + t^ 2 j $1 ± ^2j 03 ± ^1 are principal coordinates, three 

 of the times of vibration are infinite, and three equal to 



We have also 



efa 2 

 d 2 d' 



jp (0i -$?)-- h (0i - </> 2 ), to ^ 2 



Hence the three remaining principal coordinates are 



1 + <f> 2 + ^ 3 , 1 ~(ji 2 , 0! — ^, 



1 12V2 + 3 . , . 3V2 + 3, 



k 4 = j= k and fc 5 = — — k. 



4V2 + 1 W2 + 1 



Effect of the magnetic field. We can show by the method 

 already used for the tetrahedron, that the effect of the magnetic 

 field on the coordinates p, q, r is normal, that on the coordinates 

 Pi, p 2 > °i, ° 2 , r i, r 2. 4>z + ^2> $1 + 2 , 03 + ^1 half normal, and that 

 on the coordinates 0i + <p 2 + ^ 3 , 0i~ <f>2> 0i~ ^3 zero. 



Case of Seven Corpuscles. 



An arrangement in which the corpuscles are in equilibrium 

 though not as we shall see in stable equilibrium, is when six are 

 at the corners of an octahedron and one at the centre. 



If x, y, z are the coordinates of the centre of gravity of the 

 corpuscles, f>, r) r , £ r the displacement of one of the outer cor- 

 puscles relative to the central one, then we can show that x, y, z 

 are principal coordinates, and that the equilibrium is stable for 

 displacements which change only the values of x, y, z. Using the 

 nutation of the last paragraph, we find 3 + yjr 2 , fa + 2} 3 ± ik^ 

 #1 + $2 + ^3. 0i — 4>2> 0i ~ ^s, p 2 > <?2, r 2 are again principal coordi- 

 nates for whose displacements the equilibrium is stable. 



The remaining six principal coordinates are of the form 



fc+R+Mfc+fc + M-fi), 



where \, X 2 are the roots of a quadratic equation, with correspond- 

 ing coordinates for the t/'s and £"s; for these displacements the 

 equilibrium is unstable. 



The stable form for seven corpuscles would seem to be five in 



