theorem, it may be remarked, is merely laid down by the author, and the truth of it confirmed by a demonstration; it does not arise naturally in the course of the analysis; and the reader of the Mécanique Céleste is at a loss to conjecture by what train of thought it may have been originally suggested. It may be doubted whether the theorem was introduced for the sake of demonstrating a method of investigation previously known to be just from other principles; or whether it preceded in the order of invention, and led to the method of investigation. But however this may be after having studied the part of LAPLACE's work referred to with all the attention which the importance of the subject and the novelty of the analysis both conspire to excite, I cannot grant that the demonstration which he has given of his proposition is conclusive. It is defective and erroneous, because a part of the analytical expression is omitted without examination, and rejected as evanescent in all cases; whereas it is so only in particular spheroids, and not in any case on account of any thing which the author proves. Two consequences have resulted from this error; for, in the first place, the method for the attraction of spheroids, as it now stands in the Mécanique Céleste, being grounded on the theorem, is unsupported by any demonstrative proof; and, secondly, that method is represented as applicable to all spheroids differing, but little from spheres, whereas it is true of such only as have their radii expressed by functions of a particular class.

In a work of so great extent as the Mécanique Céleste, which treats of so great a variety of subjects, all of them very difficult and abstruse, it can hardly be expected that no slips nor inadvertencies have been admitted. On the other

hand, the genius of the author is so far above the ordinary cast; his knowledge of the subjects he treats is so profound; and the correctness of his views is established by so many important discoveries, that so high an authority is not to be contradicted on any material point without the greatest caution and on the best grounds. It is also to be observed that the Mécanique Céleste has now been many years before the public: and although the problem of attractions is the foundation of many important researches, and is more particularly recommended to the notice of mathematicians by the novelty and uncommon turn of the analysis; on which account it may be supposed to have been scrutinized with more than an ordinary degree of curiosity; yet nobody has hitherto called in question the accuracy of the investigation. These considerations will no doubt occasion whatever is contrary to the doctrines of LAPLACE, and more especially to his theory of the attractions of spheroids, to be received with some degree of scepticism: they ought certainly to do so; but our respect even for his authority ought not to be carried so far, as to preclude all criticism of his works, or dissent from his opinions. The writings of no author on any subject deserve to have more respect and deference paid to them, than the writings of LAPLACE on the subject of physical astronomy; with this no one can be more deeply impressed than the author of this discourse; and it was not till after much meditation that, yielding to the force of the proofs which are now to be detailed, he has ventured to advance any thing in opposition to the highest authority, in regard to mathematical and physical subjects, that is to be found in the present times.

1.* Conceive a spheroid which differs but little from a sphere, and also a point or centre in the middle; let p denote the radius of the spheroid drawn to an attracted point in the surface: then the whole spheroid will consist of two parts, viz. a sphere of which the radius is p, and a shell of matter spread over the surface of the sphere every where so thin as to contain only one molecule in the depth. The function V (which, in the law of attraction that takes place in nature, is the sum of all the molecules of the attracting body divided by their respective distances from the attracted point), relatively to the whole spheroid, will be determined by seeking its value, 1st. relatively to the sphere; 2dly, relatively to the shell of

matter.

47

3

Produce the radius p without the surfaces of the spheroid and sphere, till the distance from the centre ber; then the value of V, relatively to the sphere, for the attracted point. situate at the extremity of r, will be. † (π denoting the periphery when the diameter is unit); and, making r = p, it will be ., for the point in the surface at the extremity of p. Again, let dm be one of the indefinitely small molecules in the difference between the spheroid and the sphere; and let ƒ denote the distance of the same molecule from the attracted point in the surface at the extremity of p; then the value of V, relatively to the shell of matter spread over the surface of the sphere will be=d, the fluent being extended

3

dm

to all the molecules in the shell, those on the outside of

Méc. Céleste, Liv. 3, No. 10.

+ Liv. 2d, No. 12.

the sphere being positive, and those on the inside negative. Therefore, relatively to the whole spheroid, we shall have

3

dV

3

We must next compute the value of () in the same circumstances as before. Relatively to the sphere, it is. for the point without the surface: and, by making r = p, it is - 1. p for the point in the surface. In order to find the other part of the quantity in question we may suppose, with LAPLACE,* the attracted point to be raised up, in the prolongation of p, the distance dr above the surfaces of the spheroid and sphere; then, if f' denote the distance of the molecule dm from the attracted point in its new position, and will

dm

f

dm

ram

be two consecutive values of the same function which correspond to the values r and r+dr; therefore, supposing r to vary, the fluxional coefficient will, by the principles of the

fore, by adding together the two parts of (a).

dr

we shall get

Sam-Sam

dm

or

; (B)

observing that the second term on the right-hand side is to be valued on the supposition of dr = o.

Let y denote the cosine of the angle contained by p and another radius of the sphere drawn to the molecule dm ; then ƒ, the distance of the molecule from the attracted point in the

Liv. 3e, No. 10.

first position, will be p√2(1–7); and f', the same dis

=

tance in the second position, will be = {(p+ dr)2 — 2p(p + dr) • y + p’}1; and if, with LAPLACE, we neglect the square and

dr

other higher powers of ♪r, then ƒ' = { 1 + . }.f: there

rdm

F

Since the spheroid is supposed to approach very nearly to the spherical figure, the radius of it will fall under this form of expression, viz. p = a × (1+a =ax y); where a denotes the radius of a sphere concentric with the spheroid and nearly equal to it; a, a coefficient so small that its square and other higher powers may be neglected; and y, a function of two angles and which determine the position of p, 0 being the angle contained by p and a fixt axis passing through the centre of the spheroid, and a the angle which the plane drawn through and the axis, makes with another plane passing by the same axis. Now, by substituting and neglecting all the terms of the order a and the higher orders, the preceding values of V and () will become

and, by combining these so as to exterminatedTM, we shall

which is no other than LAPLACE's equation.*

Liv. 3, No. 10. Equation (2).