From the smallness of the mass of the ring, as well as from its unfavorable distribution, it is easy to see that r-r must be very small compared with r..

To obtain f and f', I have computed from Laplace's formula the following values: f, is the attraction of a single ring upon a particle on its surface, at the extremity of the major axis of its base; f, and f, the attractions of the two next adjacent. The interval between =0.01, 26. The radius of the outer edge of the outer ring being = 1.

=

The attraction of the whole ring, considering its mass to be uniformly distributed, I have next computed by quadratures. Breadth of whole ring 0.335. Radius of outer edge = 1.

Distance of particle within the outer edge =0.0075 Attraction +4.52X mass of ring.

66

=

These two tables give the means of finding ƒ and f' with suf ficient exactness. For Saturn we have

17/7 3

S= 40095/

; log. S=9-5567; log. mass of ring =7·4848.

The density of the ring is assumed = Saturn's, unless it be otherwise stated. A change in the density affects only that part of the ring's attraction depending on ƒ., ƒ1, and ƒ..

But f+f will be changed very nearly in the direct ratio of the different densities when the rings are narrow.

We will first suppose the case of but one ring without division.

r=0.665 r.=0.8325

-

r' = 1.000 rdr=0·335

Upon a particle at a distance within the outer edge = 0.21, the attraction of the whole ring becomes 0. This gives for the time of rotation t=0.43. The excess of Saturn's attraction over the centrifugal force at the inner edge =0·37. At the outer edge the centrifugal force is in excess by 0.33. We must therefore have,

If there be but one ring, it will be necessary to increase its attractive force by sixty times its probable value, in order to retain its particles on its surface.

With a single division into two equal rings, we have for the inner of the two, giving such a time of rotation as will retain particles on the middle from leaving their place,

As no change of mass or density within the limits of probability will account for so large differences, we must therefore still further reduce the width of the rings.

By trying different values, it will be found necessary to diminish-r so far, that the intervals occupy nearly as much area as the reflecting surface, which cannot be admitted, for reasons before given.

We will take r-r=0.02, which corresponds to eleven equal rings distant from each other by 0·01.

For the outside ring,

t=0·59

f>0·0023

ƒ'>0·0202

f=-0.0036, tendency is from the surface.
f= 0.0144

r'-r<0·0097 computed.

r-r=0·0200 assumed.

In order to preserve the mass as previously adopted, we must suppose an average density about three times that of Saturn. By recomputing ƒ and f' for the inner ring with a density =3, we obtain,

f=+0.0263

f+0.0091

r'-r<0·0101

r-r00200

A density six times that of Saturn would just suffice to retain the particles on the surface of the inner ring. To effect this without changing the mass, we must diminish b in the same proportion. But the attraction of a thin and narrow ring upon a particle at the extremity of its major axis varies nearly as bxdensity. Mécanique Céleste, vol. ii, [2095]. Therefore ƒ is not increased when we increase the density by diminishing b.

If a further diminution of width is attempted, a difficulty is encountered in the width of the intervals.

In the last case, suppose the area occupied by the intervals is already double the limit previously assigned. If we lessen the space occupied by the intervals, by bringing the adjacent rings nearer together, ƒ decreases instead of increases.

But there are still stronger objections to a large number of small rings near to each other.

It is known in the case of a single ring, that, if it were perfectly uniform in every part of its circumference, the slightest exterior disturbance would precipitate it upon the body of the planet. To avoid this catastrophe, we must suppose each ring to be an irregular solid, its center of gravity not coinciding with its center of figure, but having a motion of rotation about the body of Saturn. In addition to this, a number of regular concentric rings are in a position of unstable equilibrium, by virtue of their own mutual attractions. The slightest inequality in the intervals would have the effect of throwing the whole system into confusion.

Let us suppose, for instance, that the inner ring deviate by ever so small an amount from an exact central position with ref

erence to the ring outside it. The nearest sides commence moving together, until they come in contact. All the others must follow. The consequence of such a conflict of these masses, each urged by different velocities, corresponding to the different times of rotation of the several rings, must be fatal to the whole

structure.

It is therefore again necessary that the rings be not of regular figure or density.

But if these irregularities are small, there will be only a feeble resistance opposed to their tendency to fall upon the body of the planet.

On the other hand, if they be large, they will become the source of mutual disturbances, which must end in their destruction, by causing them to fall upon each other. The smallness of the intervals between them, and the near equality in the period of rotation of two adjacent rings, will make the danger of the latter event imminent, if not wholly unavoidable.

The nearness of the rings will in any case render it impossible that they can assume a figure of equilibrium permanent or nearly so.

The hypothesis that the whole ring is in a fluid state, or at least does not cohere strongly, presents fewer difficulties.

There being no longer an unyielding coherence between the particles of the inner and outer edges, they have not necessarily the same period of rotation about Saturn. A continual flow of the inner particles past the outer may be supposed, by which the centrifugal force will be brought into equilibrium with the other forces. And even should an accumulation of disturbances, of which the absence of inequalities lessens the probability, bring the rings together, the velocities at the point of contact will be very nearly equal, and the two will coalesce without disastrous consequences.

If in its normal condition the ring has but one division, as is commonly seen, under peculiar circumstances it might be anticipated that the preservation of their equilibrium would require a separation in some regions of either the inner or outer ring; this would explain the fact of occasional subdivisions being seen. Their being visible for a short time, and then disappearing, to the most powerful telescopes, is accounted for by the removal of the sources of disturbance, when the parts thrown off would reunite.

Finally, a fluid ring, symmetrical in its dimensions, is not of necessity in a state of unstable equilibrium with reference either to Saturn or to the other rings.

SECOND SERIES, Vol. XII, No. 34.-July, 1851.

14

ART. XV.-On the Constitution of Saturn's Ring; by Prof.
B. PEIRCE of Harvard University.

Abstract of a paper read before the American Association for the Advancement of
Science, at the meeting at Cincinnati.

A MEMOIR Upon Saturn's Ring, by Mr. George P. Bond, was read to the American Academy of Arts and Sciences, upon the 15th of April, and was the occasion of the present investigation. Since Mr. Bond's paper is still unpublished, I shall be obliged to make constant reference to it, and even recapitulate some parts of it, in order that the proper relation of the two paths of research may be correctly understood.

1. The author of the Mecanique Celeste proved that Saturn's ring, regarded as solid, would not be sustained about the pri mary, unless it had decided irregularities in its structure. But the observations of Herschel and others have failed to detect any indications of such irregularity, and a laborious series of observations have finally convinced Mr. Bond of the utter improbability of any important irregularities, and he has, therefore, adopted the conclusion that Saturn's ring is not solid but fluid. Mr. Bond's argument is chiefly derived from observation; whereas a new investigation of the mechanical conditions of the problem has led me one step further. I am now convinced that there is no conceivable form of irregularity and no combination of irregularities, consistent with an actual ring, which would permit the ring to be permanently maintained by the primary if it were solid. Hence it follows, independently of observation, that Saturn's ring is not solid. And now it is worthy of remark that if we adopt, as the basis of calculation, the mass of the ring which was determined by Bessel, the thickness from Bond and the other dimensions from Struve, we shall find the density to be about one-fourth more than water. So that the ring consists of a stream, or of streams of a fluid rather denser than water, flowing around the primary.

2. Mr. Bond next undertook a series of very ingenious and novel computations, in order to determine from theoretical considerations alone whether the ring was one or many, and arrived at the remarkable result that neither of these hypotheses was tenable. He is, therefore, disposed to reconcile the discrepancies of observation in this respect by supposing the constitution of the ring to be variable; and that, although the principal division, which has been always observed, is permanent, the other divisions are constantly annihilated by the mutual concussions of the rings, and again reproduced by some process which he does not undertake to define. This bold theory is fully sustained by my own analytical investigations, and not only do my researches ex