be determined. But after the introduction of u, the function V has the property of satisfying a partial differential equation of the second order, and by an application of the Calculus of Variations it will be proved in the sequel that the required value of V may always be obtained by merely satisfying this equation, and certain other simple conditions when p' is equal to the product of two factors, one of which may be any rational and entire function of the s accented variables, the remaining one being a simple algebraic function whose form continues unchanged, whatever that of the first factor may be.

The chief object of the present paper is to resolve the problem in the more extended signification which we have endeavoured to explain in the preceding paragraph, and, as is by no means unusual, the simplicity of the conclusions corresponds with the generality of the method employed in obtaining them. For when we introduce other variables connected with the original ones by the most simple relations, the rational and entire factor in p' still remains rational and entire of the same degree, and may under its altered form be expanded in a series of a finite number of similar quantities, to each of which there corresponds a term in V, expressed by the product of two factors; the first being a rational and entire function of s of the new variables entering into V, and the second a function of the remaining new variable h, whose differential coefficient is an algebraic quantity. Moreover the first is immediately deducible from the corresponding part of p' without calculation.

The solution of the problem in its extended signification being thus completed, no difficulties can arise in applying it to particular cases. We have therefore on the present occasion given two applications only. In the first, which relates to the attractions of ellipsoids, both the interior and exterior ones are comprised in a common formula agreeably to a preceding observation, and the discontinuity before noticed falls upon one of the independent variables, in functions of which both these attractions are expressed; this variable being constantly equal to zero so long as the attracted point p remains within the ellipsoid, but becoming equal to a determinate function of the co-ordinates of p, when p is situated in the exterior space. Instead too of seek

ing directly the value of V, all its differentials have first been deduced, and thence the value of V obtained by integration. This slight modification has been given to our method, both because it renders the determination of V in the case considered more easy, and may likewise be usefully employed in the more general one before mentioned. The other application is remarkable both on account of the simplicity of the results to which it leads, and of their analogy with those obtained by Laplace. (Méc. Cél. Liv. III. Chap. 2). In fact, it would be easy to shew that these last are only particular cases of the more general ones contained in the article now under notice.

The general solution of the partial differential equation of the second order, deducible from the seventh and three following articles of this paper, and in which the principal variable V is a function of 8+1 independent variables, is capable of being applied with advantage to various interesting physico-mathematical enquiries. Indeed the law of the distribution of heat in a body of ellipsoidal figure, and that of the motion of a nonelastic fluid over a solid obstacle of similar form, may be thence almost immediately deduced; but the length of our paper entirely precludes any thing more than an allusion to these applications on the present occasion.

1. The object of the present paper will be to exhibit certain general analytical formulae, from which may be deduced as a very particular case the values of the attractions exerted by ellipsoids upon any exterior or interior point, supposing their densities to be represented by functions of great generality.

Let us therefore begin with considering p' as a function of the s independent variables

x', x, x ......x,

and let us afterwards form the function

V =

dx'dx,'dx......dx.. p'

.....(1),

{(x, − x',')2 + (∞, − x,')2 + ... + (≈, — x,')2 + u2} ̈

the sign f serving to indicate s integrations relative to the variables x, x, x,... x,, and similar to the double and triple ones

employed in the solution of geometrical and mechanical problems. Then it is easy to perceive that the function V will satisfy the partial differential equation

seeing that in consequence of the denominator of the expression (1), every one of its elements satisfies for V to the equation (2).

To give an example of the manner in which the multiple integral is to be taken, we may conceive it to comprise all the real values both positive and negative of the variables xx,... x, which satisfy the condition

the symbol <, as is the case also in what follows, not excluding equality.

2. In order to avoid the difficulties usually attendant on integrations like those of the formula (1), it will here be convenient to notice two or three very simple properties of the function V.

In the first place, then, it is clear that the denominator of the formula (1) may always be expanded in an ascending series of the entire powers of the increments of the variables x, x,..., u, and their various products by means of Taylor's Theorem, unless we have simultaneously

x1 = x, x = x,...,x, and u = 0;

and therefore V may always be expanded in a series of like form, unless the s+1 equations immediately preceding are all satisfied for one at least of the elements of V. It is thus evident that the function V possesses the property in question, except only when the two conditions

are satisfied simultaneously, considering as we shall in what

follows the limits of the multiple integral (1) to be determined by the condition (a)*.

In like manner it is clear that when

the expansion of V in powers of u will contain none but the even powers of this variable.

Again, it is quite evident from the form of the function V that when any one of the s+1 independent variables therein contained becomes infinite, this function will vanish of itself.

3. The three foregoing properties of V combined with the equation (2) will furnish some useful results. In fact, let us consider the quantity

where the multiple integral comprises all the real values whether positive or negative of x, x,,...., with all the real and positive values of u which satisfy the condition

a a,... a, and h being positive constant quantities; and such that we may have generally

a. > a,.

In this case the multiple integral (5) will have two extreme limits, viz. one in which the conditions

* The necessity of this first property does not explicitly appear in what follows, but it must be understood in order to place the application of the method of integration by parts, in Nos. 3, 4, and 5, beyond the reach of objection. In fact, when V possesses this property, the theorems demonstrated in these Nos. are certainly correct but they are not necessarily so for every form of the function V, as will be evident from what has been shewn in the third article of my Essay On the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. [See pp. 23-27.]

1

1

are satisfied; and another defined by

x,

+ + + <1 and u= 0.

...

Moreover, for greater distinctness, we shall mark the quantities belonging to the former with two accents, and those belonging to the latter with one only.

Let us now suppose that V" is completely given, and likewise V or that portion of V' in which the condition (3) is satisfied; then if we regard V or the rest of V' as quite arbitrary, and afterwards endeavour to make the quantity (5) a minimum, we shall get in the usual way, by applying the Calculus of Variations,

seeing that SV-0 and 8V=0, because the quantities V" and

are supposed given.

The first line of the expression immediately preceding gives generally

8 dv

+ +
du u du

(2′),

which is identical with the equation (2) No. 1, and the second

line gives

av 0=u'"- 2 du

(u' being evanescent)

....(9).

From the nature of the question de minimo just resolved, there can be little doubt but that the equations (2') and (9) will suffice for the complete determination of V, where V" and V' are both given. But as the truth of this will be of consequence in what follows, we will, before proceeding farther, give a demonstration of it; and the more willingly because it is simple and very general.

4. Now since in the expression (, u is always positive,