Page:Proceedings of the Royal Society of London Vol 1.djvu/509

Rh plies it to be so. The presence of animal matter is by no means conclusive; since bones from tlle plaster quarries at Paris still contain it.

Unfortunately, our geological knowledge of Guadeloupe is yet too imperfect to assist in determining this question. The only positive information being, that the bed in which these skeletons are found is nearly an English mile in length, and that it is covered by the sea at high water.

Although it be true that three geocentric observations are really sufﬁcient for determining the parabolic orbit of a comet, as well as the elliptic orbit of a planet; the latter problem is far the easier, because we can select those positions of a planet from which its heliocentric places are found without any intricate calculation: but with regard to comets it is far otherwise. Since their appearance is unexpected, we are under the necessity of drawing our inferences from those positions in which they may happen to present themselves; and it is generally extremely difﬁcult to deduce, with accuracy, their heliocentric positions from observations necessarily conﬁned to a small part of their orbit.

In order to obtain an approximate solution, Sir Isaac Newton considered a small portion of the orbit as a straight line, the projection of which on the plane of the ecliptic will be also straight, and the parts of each will bear the same proportion to each other as the intervals of observation. But three observations alone leave the problem indeterminate; and though when four observations are employed the problem is generally determinate and easily solved, it is also often indeterminate even when four are employed.

In general it may be said that no solution is free from this imperfection, in which the velocity in the orbit does not enter as a principal condition, as in the methods of Boscovicb, Laplace, and Legendre. But in that of Laplace, the ﬁrst and second differential co-efficients of longitude and latitude can be obtained but imperfectly, and only by interpolation; and in that of Legendre his formulae are complicated, and the number of equations that require to be solved render it ill adapted for general use.

The object of the present paper is to give a new solution of the problem, which in the author's estimation, is at least as accurate as any former method; and in practice, he thinks. as commodious as the nature of such_ a calculation can well admit.

After detailing the particulars of this method, which from its nature cannot admit of abridgement, the author gives various instances of its successful application in discovering the orbits of the comets of 1769, 1781, and two comets of 1805, from observations selected by Legendre for the same purpose; and he shows, by comparison of his