SATELLITES UPON THE FORM OF SATURN’S RING. 
127 
for n — 1, and at distance 21*69" for n — 2. The last is again just beyond the limits 
of Ring A. 
The remaining satellites all produce instability at the origin, but the other points 
at which this occurs are outside the existing ring. 
We may use the observations of the dimensions of Cassini’s Division to determine 
the maximum value of F appearing. As we have already found, the inner edge 
corresponds closely to F = 0. The radius of the outer edge is a = 17'64 //f . Hence 
for satellite Mimas a/a' = 0'65781, giving k/iu = 1'0720. 
If now equation (40) be solved for F, taking this value for /c/m, the result is 
F = 0'0173. Hence we may conclude that F ranges from zero to 0*0173. 
Using this value of F, we now proceed to the study of the roots of equation (40).. 
Solving, we find 
K /m = ± F0720 and K /m = ± 2*8917. 
Take m = 1. Then 
a/a' = 0*1712 and a/a' = 0*7535. 
We may expect to find a clearance of particles from a/a' — 0 to 0*1712; and from 
a/a' = 0*7535 to unity. 
The first gives the extent of the clearance near the origin. 
For the various satellites its dimensions are : 
Mimas. 
Enceladus 
Tethys 
Dione . 
Rhea . 
Titan . 
a = 4*59", 
a = 6 * 16 ", 
a = 7* 30", 
a = 9*34", 
a = 13*07”, 
a = 29*94". 
The radius a = 9*34" indicates approximately the inner radius of the Crepe Ring,, 
while a — 13*07" indicates more closely the inner radius of Ring B. 
Applying the second ratio, a/a' = 0*7535, to Mimas, we find radius a = 20*2". 
There should be a clearance of particles from 20*2" up to the satellite itself. This 
indicates with considerable precision the termination of the whole ring, which has a 
radius 20*01". 
These results are subject to modification owing to the effect of the oblateness of 
the planet Saturn and the influence of one ring upon another. But the agreement 
of theory and observation in this first approximation is sufficiently remarkable. 
The interpretation of the effect of Dione and Rhea on the inner parts of the ring 
is not clear. From the theory one would expect that any satellite could affect a 
clearance of particles from the origin up to a radius given by a/a' = 0*1712. In that 
case Titan, the largest of the satellites, should dissipate the whole of the existing 
rings, for this ratio carries us far beyond the outer radius. 
T 2 
