still greater weight; it does not exhibit the several terms of the series for the attractive force in separate and independent expressions: it only points out in what manner they may be derived successively, one after another; in so much that the terms of the series near the beginning cannot be found without previously computing all the rest. This remark gave occasion to the following paper, in which it is my design to give a solution of the problem which is not chargeable with the imperfections just mentioned: the analysis is direct, and every term of the series for the attractive force is deduced immediately from the radius of the spheroid. As the ellipsoid, which comprehends both sorts of elliptical spheroids of revolution, falls within the class of figures here treated of, I have derived, as a corollary from my investigation, the formulas for the attractions of that figure which are required in the theory of the earth: this paper therefore will contain all that is useful on the subject of the attractions of spheroids, as far as our knowledge at present extends, deduced by one uniform mode of analysis.

Having mentioned the principal object of this discourse, I must likewise notice a subordinate purpose I have in view; it is to put in a clear light the real grounds of LaPlace's method, and of the equivalent method delivered in the following pages; to the accomplishment of which nothing is likely to contribute so much, as a direct and rigorous analysis perspicuously conducted. To promote the same end still farther, by preserving greater order and perspicuity in treating a subject in its own nature very complicated, this paper will be divided into two principal sections: in the first section it is proposed

to lay down the analytical propositions on which the investigation is founded: the second section will contain the solution of the problem under consideration.

One more preliminary observation it is proper to add. The problem of attractions contains two cases; when the density of the attracting body is uniform throughout; when it varies according to any given law: it is in the first of these two cases that the chief difficulties occur; and as I have nothing new to add on the second case, I shall here confine my attention to homogeneous spheroids, unit being supposed to denote the density.

1. Let

ре

I.

Preliminary Investigations.

denote the cosine of an angle, and let
- ƒ = { r• — 2ra. μ + aa };

then the truth of the following equation in partial fluxions will be proved merely by performing the operations indicated,

therefore, on account of the first equation, we shall obtain by

2. Let be reduced into a series of the descending powers of r; then

με

and C(1) will be a rational and integral function of of i dimensions: substitute this series for S in the equation last found n being = 0), and we shall obtain

Again, take the fluxions ʼn times successively in and like

wise in the series equivalent to it, making the only variable; and we shall get

substitute this series for S in the equation of No. 1, and we

when the fluent is taken between the limits = 1 and

the equation (1) will be included in the equation (2); whence

it is easy to infer that whatever is proved of

of the equation (2) may be transferred to C() by putting n = 0; a remark that will enable us to consult brevity, and of which we shall freely avail ourselves.

3. It is now proposed to find the value of C() in a series of the powers of .* The equation (2), by expanding its last term, will become

μ +A (s)

be assumed as equivalent to C(); then by substituting and

equating the coefficient of

A(s)

i-25 μ

to o, we shall get (i−2 s+2) (i−2 5 + 1) ̧ A(s−1);

25 (21-2 s+1)

and, by putting s=1, s=2, &c. successively, we shall hence be able to determine the proportions of all the coefficients to the first one A(), which must be investigated from other considerations. Now C) is the coefficient of a in the ex

• Méc. Céleste Liv. 3e, No. 15.

mial theorem, the term containing μ will be =

1.3.5 zi-I

2.4.6 zi

; whence it is

i(i-1) i-2 i(i—1) (i—2) (i—3)

μ

2(21-1)

+

ре

2.4.(2i—1) (2i—3)

If we take the fluxions n times successively in the last for

independent of; and when in is an odd number, the same quantity will contain a part, equal to

multiplied by μ only: these two parts of the value of

we shall afterwards have occasion to refer to.

4. It is proposed to investigate the fluent of

= 1 and μ=1; supposing P to be a

between the limits
rational and integral function of