238 Prof. J. J. Thomson on the 
due to the other (n—1) is equal to 
e” T 2a 30r (n—1) 
——, | cosec — + cosec — + cosec— +... + cosec———— 
4a n n n n 
If the corpuscles are at rest this must be equal to the radial 
attraction. Hence, if 
2 —] 
T Tv | n T 
SS, cosec — + cosec aane +... Gosec home) 
n 5 Fh n 
vera = & 9 
FT dao 
or Cie tn 
BB = Lh wt eh el Yel ae (1) 
The following are the values of S, from n=2 to n=6. 
S,=1, S;=2°3094,; S,=3°3284, 8;—5°5056, S.=-1 eee 
In the important case when v=n, 2. e. when the positive 
charge on the sphere is equal to the sum of all the negative 
charges in the ring of corpuscles, we get by (1) the following 
values for a/b :— 
a 
nN. ie 
D Virility ai. % OD 
Se rae mae “07783 
Za Neier oe 6208 
er et *6505 
(ce perreeiieree 9 °6726 
If the ring of corpuscles, instead of being at rest, is rotating 
with an angular velocity w, the condition for steady motion is 
vera e 
A 
TBE — maw? ot Aa? Sas 
3 S 
or va®> om S, 
— 2 + eet 
pee eee 
_ here m is the mass of a corpuscle. 
We shall now proceed to find the forces acting on a 
corpuscle when the corpuscles are slightly displaced from 
their. positions of .equilibrium. Let.the position. of the 
corpuscles be fixed by the polar coordinates.r and @ in the 
plane of the undisturbed orbit, and by the displacement z at 
right angles to this plane ; let 7s, 03, z; be the coordinates of 
the sth corpuscle ; then, since the corpuscles are but slightly 
displaced from their positions of equilibrium, 7,=a+ p, 
where p; is small compared with a, z, is also small compared 
with a, and 0,—6,_;= =f +4s—¢s_1, where n is the number 
of corpuscles and the ¢’s are small quantities. 

