128 
DR. G. R. GOLDSBROUGH ON THE INFLUENCE OF 
There are therefore two facts to explain. First, the existence of the Crepe Ring 
within the dissipative area of Rhea, and second, the existence of the bright rings 
within the dissipative area of Titan. In connection with the first, Lowell has noted 
a definite black band within Ring B, so that there is a clearance of particles between 
the Crepe Ring and the bright rings. It would appear as though the dissipative 
power of the satellites was only effective near the outer boundary of the unstable 
area about the origin. To discuss this, let as examine the analytical results. 
It has already been pointed out how very small the exponent c is, as given by (18), 
indicating a very slow rate of dispersion. Consider, instead, the numerators of the 
expressions (24), the vanishing of the denominators of which causes the instability. 
The numerators are small because of the quantities 0 3j))l and 0 6>jra . In the case under 
discussion, in = 1. From (5) 
e aa = /*¥*'• 
e 6a = „v 
Using the well-known expression for b u * we find 
0 3 , j = AY 4 M fa 2 + ftfa 4 +...} 
= v {w'/w— 1} “ 2 {f a 4 + fib!a” + ... } 
0<5, i = v k 2 i a (a + fa 3 + r 4 ^-a 0 ...) — a 2 } 
r r / / 11—2/3 4. 4-5 6 1 
— V 1} \-g-a +x%2‘ a ...}. 
For small values of a, w'/co is small, and the value of (w'/co— l)~ 2 will be greater than, 
but not far from, unity. Hence the values of 0 3il and 0 61 depend approximately 
upon the fourth power of a or ala'. It is clear then that the numerators in (24) will 
be vanishingly small except for the larger values of a/cd. 
The physical meaning is that, while instability will always take place when the 
denominators vanish, the rate of dissipation will be small except for the largest values 
of a which are permissible. There will also be a uniform grading in the rate of 
dissipation as a increases. 
Applying this result to the case of Saturn’s satellites, we may expect to find 
actually a clearance only near the outer limits of the areas under consideration. The 
areas of clearance of the first three satellites fall within the body of the planet. 
Dione causes the clearance between the surface of the planet at S’65" and TSI", 
which is approximately the commencement of the Crepe Ring. The limit of the area 
of clearance of Rhea is 13‘0 7", and only near that boundary is the action effective, 
the Crepe Ring being undispersed in the weaker part of the field. The bright rings 
are clearly in the weak part of Titan’s field of clearance, and so continue to exist. 
It is obvious, however, that with passage of time the Crepe Ring will be dispersed 
by Rhea and the whole by Titan. 
* Tisserand, vol. i., p. 272. 
