[ 101 ] 
IY. The Influence of Satellites upon the Form of Saturn's Ring. 
By G. R. Goldsbrougii, D.Sc., Armstrong College , Newcastle-on-Tyne. 
Communicated by Prof. T. H. Havelock, F.R.S. 
Received February 17,—Read May 26, 1921. 
§ 1. Introduction. 
In his “ Adams’ Prize Essay”'* for the year 1856, Maxwell showed that the rings 
of the planet Saturn could only be stable for small disturbances on the theory that 
they were composed of meteorites sufficiently small. This has been confirmed since 
by spectroscopic evidence and is now generally accepted. In continuance of the 
same idea, the various divisions of the rings have been accounted for by presuming 
that, in those positions where a single particle moving in a circular orbit about the 
planet would have a period simply commensurate with that of one of the nearer 
satellites of Saturn, instability would result. This idea has been fully emphasized 
recently by Lowell.! His observations at Flagstaff have disclosed a large number 
of additional divisions in the rings (see Appendix to this paper). They have the 
appearance of fine lines traced on the surface of the rings. In each case Lowell is 
able to show that the divisions occur at intervals of periods commensurable with that 
of satellite Mimas. The periods have the ratios such as §, T fi r> ^, xt, & c - Lowell 
has stated the argument for this view in ‘ Bulletin,’ 32, p. 189. If the action of one 
body upon another revolving about a third be examined by the method of the 
variation of arbitrary constants, in the expressions for the periodic inequalities 
in the radius vector and the longitude, there appear terms of the type 
[C/(pn — qn')\ cos {(pn — q?i') t + Q}, where n and n' are the mean motions of the 
perturbing and perturbed bodies, p and q are integers, and the remaining quantities 
are constants. It is clear that when the ratio n/n' is approximately equal to q/p, 
then the inequality will become very large. 
We may take a satellite of Saturn as one of the bodies and one of the particles 
forming the ring as the other ; if n/n' = q/p, approximately, then the particle will 
* Maxwell’s ‘ Collected Works,’ I., p. 288. 
f Lowell, ‘ Observatory Bulletin,’ No. 66. 
VOL. CCXXII.—A 597. Q 
[Published October 13, 1921. 
