section of a plane perpendicular to the ray Op with the surface of the sphere itself, provided n is greater than 2. When on the contrary n is less than 2, this law requires a certain modification; the nature of which has been fully investigated in the article just named, and the one immediately following.

It has before been remarked, that the generality of our analysis will enable us to assign the density of the free fluid which would be induced in a sphere by the action of exterior forces, supposing these forces are given explicitly in functions of the rectangular co-ordinates of the point of space to which they belong. But, as in the particular case in which our formulæ admit of an application to natural phenomena, the forces in question arise from electric fluid diffused in the inducing bodies, we have in the ninth article considered more especially the case of a conducting sphere acted upon by the fluid contained in any exterior bodies whatever, and have ultimately been able to exhibit the value of the induced density under a very simple form, whatever the given density of the fluid in these bodies may be.

The tenth and last article contains an application of the general method to circular planes, from which results, analogous to those formed for spheres in some of the preceding ones, are deduced; and towards the latter part, a very simple formula is given, which serves to express the value of the density of the free fluid in an infinitely thin plate, supposing it acted upon by other fluid, distributed according to any given law in its own plane. Now it is clear, that if to the general exponent n we assign the particular value 2, all our results will become applicable to electrical phenomena. In this way the density of the electric fluid on an infinitely thin circular plate, when under the influence of any electrified bodies whatever, situated in its own plane, will become known. The analytical expression which serves to represent the value of this density, is remarkable for its simplicity; and by suppressing the term due to the exterior bodies, immediately gives the density of the electric fluid on a circular conducting plate, when quite free from all extraneous action. Fortunately, the manner in which the electric fluid

distributes itself in the latter case, has long since been determined experimentally by Coulomb. We have thus had the advantage of comparing our theoretical results with those of a very accurate observer, and the differences between them are not greater than may be supposed due to the unavoidable errors of experiment, and to that which would necessarily be produced by employing plates of a finite thickness, whilst the theory supposes this thickness infinitely small. Moreover, the errors are all of. the same kind with regard to sign, as would arise from the latter cause.

1. If we conceive a fluid analogous to the electric fluid, but of which the law of the repulsion of the particles instead of being inversely as the square of the distance is inversely as some power n of the distance, and suppose p to represent the density of this fluid, so that du being an element of the volume of a body A through which it is diffused, pdv may represent the quantity contained in this element, and if afterwards we write g for the distance between du and any particle p under consideration, and these form the quantity

the integral extending over the whole volume of A, it is well known that the force with which a particle p of this fluid situate in any point of space is impelled in the direction of any line q and tending to increase this line will always be represented by

V being regarded as a function of three rectangular co-ordinates of p, one of which co-ordinates coincides with the line

I, and

(2) being the partial differential of V, relative to this last co

ordinate.

In order now to make known the principal artifices on which the success of our general method for determining the function V mainly depends, it will be convenient to begin with a very simple example.

Let us therefore suppose that the body A is a sphere, whose centre is at the origin 0 of the co-ordinates, the radius being 1; and p is such a function of x', y', z', that where we substitute for x', y', ' their values in polar co-ordinates

x' =r' cos 0, y' =r' sin e' cos a', z' =r'′ sin e′ sin ☎',

it shall reduce itself to the form

p = (1-p').f(r'3);

f being the characteristic of any rational and entire function whatever which is in fact equivalent to supposing

p = (1 − x'3 — y′′ — z")3 . ƒ (x2 + y2 + z'3).

Now, when as in the present case, p can be expanded in a series of the entire powers of the quantities x', y', z', and of the various products of these powers, the function V will always admit of a similar expansion in the entire powers and products of the quantities x, y, z, provided the point p continues within the body 4*, and as moreover V evidently depends on the distance Op=r and is independent of 0 and w, the two other polar co-ordinates of p, it is easy to see that the quantity V, when we substitute for x, y, z these values

x=r cos 0, y = r sin cosa, z = r sin 0 sin,

will become a function of r, only containing none but the even powers of this variable.

But since we have

də

dv=r" dr' dỡ do' sin, and p = (1- y'3)3. ƒ (r'3),

the value of V becomes

* The truth of this assertion will become tolerably clear, if we recollect that may be regarded as the sum of every element pdv of the body's mass divided by the (n − 1)th power of the distance of each element from the point p, supposing the density of the body A to be expressed by p, a continuous function of x, y, 2. For then the quantity V is represented by a continuous function, so long as p remains within A; but there is in general a violation of the law of continuity whenever the point p passes from the interior to the exterior space. This truth, however, as enunciated in the text, is demonstrable, but since the present paper is a long one, I have suppressed the demonstrations to save room.

=

the integrals being taken from 0 to '2′′, from '=0 to '=, and from r'=0 to r' = 1.

Now may be considered as composed of two parts, one V due to the sphere B whose centre is at the origin O, and surface passes through the point p, and another V" due to the shell S exterior to B. In order to obtain the first part, we must expand the quantity g1 in an ascending series of the powers of. In this way we get

g1TM = [r3 — 2rr′ {cos & cos 0' + sin ✪ sin e' cos (☛' — ≈)} +~3]="

If then we substitute this series for g in the value of V', and after having expanded the quantity (1 − r), we effect the integrations relative to r', ', and ', we shall have a result of the form

V' = p1” {A + Br3 + Cra+&c.}

seeing that in obtaining the part of V before represented by V', the integral relative to rought to be taken from r'=0_to_r′ =r only.

To obtain the value of V", we must expand the quantity g1 in an ascending series of the powers of, and we shall thus

the coefficients Q, Q1, Q2, &c. being the same as before.

The expansion here given being substituted in V", there will arise a series of the form

V" = T2+ T2+ T2+ T2+&c.

of which the general term T, is

зов

T‚= fd0' dw' sin 0′ Q. fr” dr' — ' - (1 − r'1)®, ƒ (r'2) ;

the integrals being taken from r'r to r1, from

=

=

=

0 to

e, and from 0 to 27. This will be evident by recollecting that the triple integral by which the value of V" is expressed, is the same as the one before given for V, except that the integration relative to r', instead of extending from r' = 0 to r'1, ought only to extend from r'=r to r = 1.

But the general term in the function f(r) being represented by Ar", the part of T, dependent on this term will evidently be Ar fd6' dw' sin Oʻ. Q. fr'243-8′′ dr′ (1 — 7'2)3.................................. (2) ;

the limits of the integrals being the same as before.

We thus see that the value of T, and consequently of 17" would immediately be obtained, provided we had the value of the general integral

but since the first line of this expression is the well-known expansion of

when n=2.p=b+1 and q = 2 (B+1) we have ultimately,

By means of the result here obtained, we shall readily find the value of the expression (2), which will evidently contain one term multiplied by and an infinite number of others, in all of