124 
DR. G. R. GOLDSBROUGH ON THE INFLUENCE OF 
We shall assume that the number of particles in any ring is large. It is probable 
that they vary in magnitude from the infinitesimally small up to the limit given by 
Maxwell. Hence the values of the expressions F x , Gyx, H\ and Jfi will vary over 
a range of values, between the given limits, as X takes its successive values. 
Reverting to determinant (36), we see 
that it may 
now be written 
m 2 +0] 
2 p 
o ? K ' J r 2i i , 
kTTs.j,. 
K' 2 G 
— 2 Kim , 
0 
0 ,.. 
., 0 
0 
0 
o ; 
-m 2 + Q l 0 , 
M'xi , 
•; * J'/M 
K J G ], 2 
5 — wr + Gj.o, 
GG 3 '2, • 
., k 2 G „, 2 ; 
0 
— 2 Kim , 
0 ,.. 
., 0 
0 
, 2 Kim , 
o 
o ; 
2T' 
K 1,2 ) 
-m 2 +ei 0 , 
k 2 J' 3 , 2 ,.. 
2T' 
• 5 K ° >1,2 
aW, „ 
> k 2 G 2 , „ 
K a j • 
•, —nr + 0i >o j 
0 
0 
0 
. , — 2 Kim 
0 
, 0 
0 ,. 
., 2 Kim ; 
^ 2 'Fl,n , 
ic 2 J' 2 , n > 
K 2 'Ill” 
— ?U 3 + 05, o 
J 
For all conditions satisfying Maxwell’s criterion, the quantities G Kix , J\ ifl will be 
small. So that, provided k 2 is not too great, the value of the determinant (37) will 
be small for those values of k that satisfy the relation 
—m 2 + 0] )O , 
0 
0 
— 2 Kim , 
0 
0 
= 0 ' 
2 Kim, , 
0 
0 
ill? 4 0(o, 
0 
0 
0 
— m 2 + 0f_ o, . 
0 
0 
— 2 Kim , . 
0 
0 
2iam , . 
0 
0 
-m 2 + Gl 0 , . 
0 
0 
0 
.., -m 2 + 9G, 
0 
0 
... 2 Kim 
0 
0 
.. , 2 Kim , 
0 
0 
- m 2 +0’l tl 
> 
This relation is satisfied by those values of k which satisfy the equation 
(— m 2 + 0\ o ) (— m 2 + 0 x 5jO ) — 4«: 2 m- = 0, ...... (39) 
where X takes all its integral values in turn. Further it is easily shown that, on any 
distribution with n large, — F A = TH\. Hence we fall back upon the same type 
of equation as we had in the case of equal particles (equation (29)) where we replace 
vL s by F x . 
Instead of treating the equation (39) separately for the various integral values 
of X, since n is large, we may imagine a single equation with the assumption that 
F A is an arbitrary variable parameter. The determinant (37) will then be small, and 
instability result for all the values of k given by (39), for all values of the parameter 
F x that exist. With a wide range of values of F A corresponding to a wide range in 
