126 
DR. G. R. GOLDSBROUGH ON THE INFLUENCE OF 
vh s or F. 
«/ 
m. 
m 
= 1 . 
m = 2. 
a l 
a’. 
a/a'. 
0 
1 
CO 
0 
1 
0-6299 
o-oi 
1-0395 
3-9336 
0-0243 
0-8223 
0 ■ 6456 
0-015 
1-0639 
3-1412 
0-1534 
0-7745 
0-6549 
0-020 
1-0910 
2-6509 
0-2102 
0-7349 
0-6656 
0-030 
1-1750 
2-0156 
0-2810 
0-6333 
0-038 
1-3353 
1-5772 
0-3981 
0-5117 
0-039 
1-4424 
— 
0•4549 
Remembering that k — —w («/—o>) *, w 2 a 3 = w' 2 a 3 , we find : 
For m — 1, co = oo, a/a' - 0, 
m = 2, o)/(o' = 2 , a/a' = CT62996, 
m = 3, »/«' - | , a/a' = 076289, 
m = 4, co/a/ = -f- , a fa' — 0‘82524. 
[It should be remarked that a positive value of k gives positions without the 
satellite orbit, and a negative value of k gives positions within. As k appears in (40) 
in the form of a square, both positive and negative forms result. We should 
therefore have the same phenomena in a ring of particles beyond the satellite orbit 
as we find within]. 
The result a = 0 implies a division of the ring at the origin. This would fall 
within the planet itself. But if the zone consequent upon the variation of F is 
extensive, it may extend beyond the surface of the planet and show a clearance 
of particles there. 
For m = 2, a/a' = 0'62996. In the case of satellite Mimas this should indicate 
the commencement of a division in the ring at distance 1 6 '9 /r . Cassini’s Division 
begins at lhTZ" and ends at 17'64'h This agreement is very remarkable. 
Reference to the figure shows that in the vicinity of /c/m = 1, #c increases with F. 
But as k increases so does a)a’. Hence the instability caused by the larger values 
of F should be in positions corresponding to larger values of k, that is, to larger values 
of aja'. In other words, the division should extend outwards. This agrees with the 
observational data just quoted. We may then attribute the production of Cassini’s 
Division to Mimas. 
For n = 3, a/a' = 076289. 
For satellite Mimas, this should cause a division at distance 20'46 // . This is just 
beyond the outer edge of Ring A, which terminates at 20'OlA 
Considering next the satellite Enceladus, we should find a division at the origin 
