SATELLITES UPON THE FORM OF SATURN’S RING. 
119 
where v is the ratio of the mass of a particle to the mass of Saturn and p is the 
number of particles in the ring, must he fulfilled. 
v appears in the expressions for 0 1O and 0 5iO in the form i/L s . It has been 
mentioned that L, < 0’01S4n 3 for all values of ,s\ Hence iL s < 0'0194m 3 . For the 
present we shall regard vL s as a variable parameter and discuss the solutions relative 
to this parameter. 
In order to locate the zone of instability, we equate expression (27) to zero. 
Writing it in full, but omitting the term involving /, which will be exceedingly 
small and will hardly affect the result, we find 
{(•3 — vlu s ) k 2 p7i 2 } {2kL s ./ c 2 + 7 ? 2 }— 4:K 2 n 2 — 0.(29) 
This equation, regarded as involving an unknown quantity n 2 //c", is precisely the 
equation used by Maxwell to determine the condition of stability of the ring of 
particles when unperturbed by any satellite. The condition of the reality of n 2 /ic 2 
leads to the upper limit for v just quoted. In our problem we may take the unknown 
quantity as K 2 /n 2 , and then assuming a value for vL s , solve the equation. The values 
of k (for differing values of n) will give the position of the zones of instability of a 
ring of particles of mass and number assumed. Or, conversely, taking a position of 
instability, as shown by telescopic observations of the ring, we may determine the 
corresponding value of jfL s , which establishes the order of value of the mass and 
number of particles at that point. 
I have found that the latter process leads to no satisfactory result, and hence I do 
not record the work. 
It is interesting to examine the meaning of the condition previously referred to, 
that the maximum instability is found when (cq, o 05 ,o — n*) is approximately zero. On 
referring again to equation (28), it is clear that the broadest zone of instability 
will be found, owing to the extreme smallness of the last member, when 
(0 liO — w 2 ) (0 5 ,o — n 2 ) — 4 k ii 2 changes most slowly with k. This will occur when the 
equation (29) has equal roots. Equal roots appear when, by the variation of the 
parameter kL s , k/h passes from real to imaginary values, or when* 
vL s = 0-039. 
This is the upper limit of the criterion previously quoted from Maxwell, and 
would imply that all the particles were of such mass and number as to be on the 
border-line of instability. 
When i/L s has this value, we find that 
2i'L s (3 — iTj 5 ) k 1 — n* = 0 ; 
or 
05 , 001,0 —^ = 9 . 
* Tisserand, loc. cit., p. 183. 
S 2 
