SATELLITES UPON THE FORM OF SATURN’S RING. 
125 
the magnitudes of the masses of the particles, we may expect to find a broad region 
of instability. 
It can readily be shown that the condition (39) would also be produced if the 
general case of unequal particles were solved for the complementary function in the 
same way as has been done for the case of equal particles, which produced (29). 
This work is not reproduced owing to the length and complexity of the expressions, 
and also because the results are wholly contained in the condition (39) produced from 
the particular integral. 
§ 6. Application of the Results to the Saturnian System. 
Equation (39) written out in full is 
{/c 2 (3 —F A )-bm 2 } {2F A /c 2 + m 2 }-4/cW = 0. . . . 
(40) 
In this equation m is any integer and F A may vary between zero and 0'0096n 3 m/M. 
As the distinctions indicated by the suffix A are 
now of no importance, it may be dropped. The 
solutions of (40) will give approximately the 
positions where divisions in the Ring of Saturn 
may be expected. 
For any given value of F, there are four 
values of /c/m, two pairs equal with opposite signs. 
For any given value of /c/m there are two values 
of F; one, however, being greater than the 
Maxwell limit, is excluded. The limiting value 
of F for real values of /c/m is 0'039. This is, 
of course, the same result as that found by 
Maxwell. 
The relation between /c/m and F is shown in 
the figure, and the table shows actual numerical 
values. 
We may readily assume that in the existing 
Rings of Saturn there are particles of all masses 
from the infinitesimal to Maxwell’s upper limit. 
These will give rise to varying values of F, 
depending upon the masses of the particles 
adjacent to the particle under consideration. The 
maximum value of F is itself small compared with unity ; we shall then arrive at a 
limit of /c by taking F = 0 in equation (40). We find thus that the boundary of a 
division should occur at /c/m = 1, for each integral value of"m. 
VOL. CCXXII.—A. T 
Values of vL s or A 
