the Tores of Saturn. 477 
inasmuch as there are many particles in any cross-section of 
the ring, there must be a component of motion in any collision 
tending to throw the colliding particles out of the plane of 
the ring, either above or below it. Such extra-plane particles 
would, therefore, be most numerous just inside the points of 
commensurability, because, though the moment of momentum 
is preserved and thus particles be thrown outward from the 
point as well as in, owing to the loss of energy they must be 
more numerous on the inside, 
Considering, now, the commensurate ratios between the 
periods of particle and satellite which can enter into the 
problem, we find these in the order of their potency to be : 
With Mimas, 
j- 
• 2 
1 : 
:3, 
1 : 
:*, 
With Enceladus, 
1 
:3, 
With Tethys, 
1 
: 4. 
Such periods of commensurability as 2 : 3 of Mimas and 
1 : 2, 2 : 3 of Enceladus do not come into question as they 
take place outside the ring-system. Now calculation shows 
that the distances corresponding to a period of 1 : 2 of Mimas, 
of 1 : 3 of Enceladus, and of 1 : 4 of Tethys fall in Cassini's 
division, which separates ring A from ring B. The first or 
outer tore should therefore occur just inside that division 
or in the outer part of ring B. This is precisely where we 
find it ; for the inner edge of Cassini's division is at 1* 92 radii 
of Saturn from the centre of the planet, and the outer tore 
begins at 1*92, thence to stretch inward toward the disk. 
Pursuing our inquiry with the next most effective ratio, that 
of 1:3 of Mimas' period, we note that its corresponding 
distance falls at the boundary of ring B and ring C at 1*495 
radii of Saturn from the centre. Now it is inside this, to wit, at 
1*46 and 1'42, that the inner tore begins. Furthermore, this 
tore is much longer than the outer one. We turn, therefore, 
to the next most potent ratio, that of 1 : 4 of Mimas' period, 
to find that its distance falls at 1*24. This then accounts 
for the greater length of the inner tore. 
The remarkable way in which theory thus accounts for 
observation is of interest, and the more so from involving a 
case of celestial mechanics interesting in itself. 
Phil. Mag. S. 6. Vol. 15. No. 88. April 1908. 2 K 
