102 
DR. G. R. GOLDSBROUGH ON THE INFLUENCE OF 
depart considerably from its unperturbed path and collision with other particles will 
result. In this way the divisions in the ring have been explained. 
Some doubt has been cast upon this theory, and it has been shown* that even 
when n and v! are commensurable, a closer examination of the motion leads to the 
conclusion that the denominator will not vanish. 
It is also noticeable that this explanation takes no account of the attraction of the 
numerous particles upon one another, which may be considerable. 
A re-examination of the matter is made in the present paper. As the satellites of 
Saturn are all approximately in the same plane as the ring, the problem is formulated 
in two dimensions only. The satellite is assumed to follow an unperturbed circular 
orbit, and the problem reduces to a slight variation of the “ restricted problem ” of 
three bodies. We shall consider the effect of this satellite upon a number of particles 
forming a single ring round the planet, subject to their mutual attraction as well as 
that of the satellite and of Saturn. The actual Saturnian rings are supposed to be 
composed of a number of such rings arranged concentrically. These will have some 
effect one upon the other, but, for the present, this effect is disregarded. 
In his paper, Maxwell considered the single ring of particles only. He found 
that the equations of motion could be satisfied by assuming that the particles rotated 
round the primary in a circle with suitable angular motion. He then examined the 
effect of a small arbitrary disturbance upon them, and his results show that the 
disturbances would remain small if the masses of the particles were sufficiently small. 
That is, the ring would be “ ordinarily ” stable. 
In the present paper the plan is different. The disturbance of the ring of particles 
by the satellite is examined, with a view to determining under what conditions the 
departure from a certain fixed circle will be large. It is clear that if the departures 
do become large, collisions with adjacent rings of particles will result, and the particles 
will leave the vicinity of the original circle irrevocably. In this case a division in 
the ring will result. It is with this meaning that the terms stability and instability 
have been used in the paper. But-, as will be pointed out again in its proper place, 
the orbits in which the departure from the circular form does not become great with 
increase of time may yet become “ ordinarily ” unstable if further small arbitrary 
displacements are imposed upon them. 
The results of this paper will therefore indicate some, but not necessarily all, of the 
positions of divisions in the rings due to instability of whatever kind. 
In §§ 2 to 4 an analytical theory is fully worked out on the supposition of equal 
particles in each ring. In § 5 it is shown how amendments may be introduced to 
cover the case of unequal particles. The application to the Saturnian system is given 
in § 6, and the last paragraph summarises the results obtained. 
* Tisserand, ‘ Mec. Celeste,’ vol. iv., >p. 420. 
