124 ON THE ORBITS OF THE ASTEROIDS. 
origin of the asteroids; one, that they are the fragments of a single planet, which was 
shattered by the operation of some unknown cause; another, that they were formed 
by the breaking up of a revolving ring of nebulous matter. 
The first of these is the celebrated hypothesis of Olbers. It has the advantage of 
accounting for the phenomena, considered in their more salient aspects, in a very re- 
markable manner. The normal solar system is still the solar system as we should 
expect it to be in the absence of any knowledge of the planet or planets revolving be- 
tween Jupiter and Saturn. The phenomenon (abnormal on this hypothesis) of the 
place of a single large planet being filled by a collection of small ones of varying 
brilliancy, large inclinations and eccentricities, and slightly different mean distances, is 
precisely what might be expected as the result of a force which should break the 
planet into fragments, each very small in comparison with the original planet. But 
we shall see hereafter that when we carry the results of this hypothesis to numerical 
exactness, the observed phenomena are very far from agreeing with these results. 
Moreover, it is difficult, perhaps impossible, to imagine how any known natural cause, 
or combination of causes, should produce such a result as the shattering of a planet. 
But since the limits of our knowledge are not necessarily the limits of possibility, this 
objection is not fatal, and it is difficult to say what weight ought to be assigned to it. 
The second hypothesis, that the asteroids were formed by the breaking up of a ring 
of nebulous matter, is not at all improbable if the nebular hypothesis is true, but it is 
subjected to most or all the uncertainties of that hypothesis. The ring must have 
been considerably inclined to the plane of the ecliptic, or none of its fragments could 
have fallen into orbits much inclined to that plane; and it could not have revolved in 
its own plane, else all the fragments would have had nearly the same inclination to the 
invariable plane of the solar system; and it must have been somewhat eccentric, else 
all the fragments would have had about the same mean distance. Now it is remarka- 
ble that the two last circumstances would cause a tendency in the ring to break into 
fragments, while a circular ring, revolving in its own plane, would have no such ten- 
dency. We should then expect, in case a ring should break up, that its fragments 
would present some at least of the phenomena presented by the asteroids. But the 
hypothesis is not equally susceptible with that of Olbers of a posteriori tests. 
To apply rigorous tests to either of the above-mentioned hypotheses, we need rigor- 
ous expressions in terms of the time for the values of the eccentricity, inclination, lon- 
gitude of perihelion, and longitude of node, of each asteroid used in applying the test. 
For the probable tests we need the mean and the limiting values of the same elements, 
and to obtain these the same expressions are necessary. ` ` 
