containing 4, 5, 6, 7, and 8 corpuscles, etc. 47 



one plane at the corners of a pentagon, the other two on opposite 

 sides of the plane, the line joining them passing perpendicularly 

 to the plane through the centre of the pentagon, the distance of 

 either of these from the centre is about ^ of the distance of the 

 other five from the same point. 



Case of Eight Corpuscles. 



These when in equilibrium will be arranged at the corners of 

 a cube. Take as the axes of x, y, z the lines through the centre 

 of this cube parallel to its sides. Denoting the eight corpuscles 

 by the symbols (1), (2), (3), (4), (5), (6), (7), (8), let the coordi- 

 nates of these when in equilibrium be given by the scheme 



(1) (a, a, a), (2) (— a, a, a), (3) (a, — a, a), (4) (a, a, — a), 



(8) (— a, — a,- a), (7) (a, — a, a), (6) (— a, a, — a), (5) (— a, —a, a). 



Let the displacement of the rth particle from its position of 

 equilibrium be £ r , r) r , £,.. 



For the sake of brevity denote £ r + £ 9 _ r by X r , % r — £ 9 _ r by X r ', 

 Vr + Vv-r by Yr, Vr - Vs~r by F/, £ r + £ 9 _ r by Z r , £ r -£V_ r by Z r \ 

 then we may show that 



m ~(X 1 + X 2 + X 3 + X 4 ) = -k(X 1 + X 2 + X 3 + X i ), 

 m ^(Y l+ Y 2 + Y 3 +Y i ) = -k(Y 1 + Y 2 + F..+ F 4 ), 



m~(Z 1 + Z 2 + Z 3 + Z 4 ) = -k(Z 1 + Z 2 + Z 3 + Z i ), 



8e 2 

 where k = -jj . an d b is the radius of the sphere of positive 



electrification. Thus SX, %Y, 2^ are principal coordinates, they 

 are proportional to the coordinates of the centre of gravity; the 



I TYh 



time of vibration corresponding to these coordinates is 27r . / ~r- 



If X 1 + X 3 - (X 2 + X 4 ) + Y,+ Y 2 -(Y 3 +Y 4 ) = ^, 

 Z 1 + X 3 -(Z 2 + X 4 )-(F 1 +F 2 )+F 3 +F 4 =^ 2 , 

 X 1 + X i -{X 2 + X 3 ) + Z l + Z 2 -(Z 3 + Z i ) = <f> 1 , 

 X, + X 4 - (X 2 + X 3 )-(Z l + Z 2 ) + Z 3 + Z 4 = fa , 

 Y 1 +Y i -(Y 2 +Y 3 ) + (Z 1 + Z 3 )-(Z 2 + Z 4 ) = d l , 

 Y 1 +Y i -(Y 2 +Y 3 )-(Z 1 + Z. 3 ) + (Z 2 + Z i ) = d 2 , 



