the particular value 4 being one half only of what would result from making i=0 in this general formula.

But e0 evidently gives E= 0, and therefore the expansion of ƒ (x', y') before given becomes

ƒ (x', y') = A' + Aa2 cos &+4" cos 20′ + A cos 30'+&c.

or by summing the series included between the braces,

R being the distance between P, the point in which the quantity of fluid q is concentrated, and that to which the density p is supposed to belong.

Having thus the value of ƒ (x', y') we thence deduce

The value of p here given being expressed in quantities perfectly independent of the situation of the axis from which the angle e' is measured, is evidently applicable when the point P is not situated upon this axis, and in order to have the complete value of p, it will now only be requisite to add the term due to the arbitrary constant quantity on the left side of the equation (26), and as it is clear from what has preceded, that the term in question is of the form

const. × (1 — 7'),

we shall therefore have generally, wherever P may be placed,

The transition from this particular case to the more general one, originally proposed is almost immediate: for if p represents the density of the inducing fluid on any element do, of the plane coinciding with that of the plate, p,do, will be the quantity of fluid contained in this element, and the density induced thereby will be had from the last formula, by changing g into p,do, If then we integrate the expression thus obtained, and extend the integral over all the fluid acting on the plate, we shall have for the required value of p

R being the distance of the element do, from the point to which p belongs, and a the distance between do, and the center of the conducting plate.

Hitherto the radius of the circular plate has been taken as the unit of distance, but if we employ any other unit, and sup

pose that b is the measure of the same radius, in this case we

do, and R respectively, recollecting that is a quantity of the

P1

.

dimension 0 with regard to space, by so doing the resulting

By supposing n=2, the preceding investigation will be applicable to the electric fluid, and the value of the density induced upon an infinitely thin conducting plate by the action of a quantity of this fluid, distributed in any way at will in the plane of the plate itself will be immediately given. In fact, when n= 2, the foregoing value of p becomes

If we suppose the plate free from all extraneous action, we shall simply have to make p=0. in the preceding formula; and thus

Biot (Traité de Physique, Tom. II. p. 277), has related the results of some experiments made by Coulomb on the distribution of the electric fluid when in equilibrium upon a plate of copper 10 inches in diameter, but of which the thickness is not specified. If we conceive this thickness to be very small compared with the diameter of the plate, which was undoubtedly the case, the formula just found ought to be applicable to it, provided we except those parts of the plate which are in the immediate vicinity of its exterior edge. As the comparison of any results mathematically deduced from the received theory of electricity with those of the experiments of so accurate an observer as Coulomb must always be interesting, we will here give

a table of the values of the density at different points on the surface of the plate, calculated by means of the formula (29), together with the corresponding values found from experiment.

We thus see that the differences between the calculated and observed densities are trifling; and moreover, that the observed are all something smaller than the calculated ones, which it is evident ought to be the case, since the latter have been determined by considering the thickness of the plate as infinitely small, and consequently they will be somewhat greater than when this thickness is a finite quantity, as it necessarily was in Coulomb's experiments.

It has already been remarked that the method given in the second article is applicable to any ellipsoid whatever, whose axes are a, b, c. In fact, if we suppose that x, y, z are the co-ordinates of a point p within it, and x', y', z' those of any element dv of its volume, and afterwards make

x = a.cos 0, y:

=

b. sin cos ;

z = c. sin @ sin ☎,

'= c. sin e' sin ',

x =
'= a.cose, y' = b. sin e' cos a',

we shall readily obtain by substitution,

V = abe sp. r'adr'de'dw' sin 0′. (λr2 — 2μrr' + vr'2)=;

the limits of the integrals being the same as before (Art. 2), and

λ= a2 cos 02+b2 sin cos

μ=a2cose cose+b2 sin◊ sin

v=a2 cos 02+b2 sin e' cos

+ c2 sin o sin 3,

cosa cos '+c'sine sin o'sina sina', 2+ c2 sin e' sin ".

12

V=

Under the present form it is clear the determination of V can offer no difficulties after what has been shown (Art. 2). I shall not therefore insist upon it here more particularly, as it is my intention in a future paper to give a general and purely analytical method of finding the value of V, whether p is situated within the ellipsoid or not. I shall therefore only observe, that for the particular value

the series U+U' + U' + &c. (Art. 2) will reduce itself to the single term U., and we shall ultimately get

ƒ ̃de′sin0′f;” ̃da' (a2cos 0'2+b'sin @cos "+c'sin 'sino')",

which is evidently a constant quantity. Hence it follows that the expression (30) gives the value of p when the fluid is in equilibrium within the ellipsoid, and free from all extraneous action. Moreover, this value is subject, when n < 2, to modifications similar to those of the analogous value for the sphere (Art. 7).