But dy' and dV' being perfect differentials, X'dx' + Y'dy' + Z'dz must be so likewise, making therefore

do' = X'dx' + Y'dy' + Z'dz',

the above, by integration, becomes

=

const. (1k) $' + ky' + kV'.

Although the value of k depends wholly on the nature of the body under consideration, and is to be determined for each by experiment, we may yet assign the limits between which it must fall. For we have, in this theory, supposed the body composed of conducting particles, separated by intervals absolutely impervious to the magnetic fluid; it is therefore clear the magnetic state induced in the infinitely small sphere de', cannot be greater than that which would be induced, supposing it one continuous conducting mass, but may be made less in any proportion, at will, by augmenting the non-conducting intervals.

When du' is a continuous conductor, it is easy to see the value of the potential function at the point p", arising from the magnetic state induced in it by the action of the forces X, Y, Z, will be

3dv X (x′′ − x') + Y (y′′ − y') + Z (z2 − z')

4π

seeing that

3dv
4π

18

a2; a representing, as before, the radius of the sphere dv'. By comparing this expression with that before found, when do' was not a continuous conductor, it is evident k must be between the limits 0 and π, or, which is the same thing,

g being any positive quantity less than 1.

The value of k, just found, being substituted in the equation serving to determine p', there arises

Moreover

— z)

*'- fdz dy de X (x' — 2) + Y (y' − y) + Z (s' − s)

=

dz

the triple integrals extending over the whole volume of A, and that relative to do over its surface, of which do is an element;

and consequently &p=0; the symbol & referring to x', y', z' the co-ordinates of p'; or, since a', y' and ' are arbitrary, by making them equal to x, y, z respectively, there results

r being the distance p', do, and belonging to do. The

former equation serving to determine d' gives, by changing

do

dw

x', y', z' into x, y, %,

const. = (1−9) p + 32 (↓ + V)

(c);

4, and belonging to a point p, within the body, whose coordinates are x, y, z. It is moreover evident from what precedes that the functions, and V satisfy the equations 0=&p, 0=84 and 0 = 8, and have no singular values in the interior of A.

The equations (b) and (c) serve to determine p and ý, completely, when the value of V arising from the exterior bodies is known, and therefore they enable us to assign the magnetic state of every part of the body A, secing that it depends on X, Y, Z, the differential co-efficients of p. It is also evident that y', when calculated for any point p', not contained within the body A, is the value of the potential function at this point arising from the magnetic state induced in A, and therefore this function is always given by the equation (b).

The constant quantity g, which enters into our formulæ, depends on the nature of the body solely, and, in a subsequent article, its value is determined for a cylindric wire used by Coulomb. This value differs very little from unity: supposing therefore g=1, the equations (b) and (c) become

do " dw

const. =+ V

evidently the same, in effect, as would be obtained by considering the magnetic fluid at liberty to move from one part of the conducting body to another; the density p being here replaced by and since the value of the potential function for any point exterior to the body is, on either supposition, given by the formula (b), the exterior actions will be precisely the same in both cases. Hence, when we employ iron, nickel, or similar bodies, in which the value of g is nearly equal to 1, the observed phenomena will differ little from those produced on the latter hypothesis, except when one of their dimensions is very small compared with the others, in which case the results of the two hypotheses differ widely, as will be seen in some of the applications which follow.

If the magnetic particles composing the body were not perfect conductors, but indued with a coercive force, it is clear there might always be equilibrium, provided the magnetic state of the element du' was such as would be induced by the forces dV' dy dv

dy

+ + A',
dx'

đi đi anh

dự dV +

dx dx'

+ + B' and

dy'

dy'

dy dy

d . dy
dz' + dž

+C', instead of

; supposing the resultant

of the forces A', B', C' no where exceeds a quantity B, serving to measure the coercive force. This is expressed by the condition

A+B+C" <B2,

the equation (c) would then be replaced by

A, B, C being any functions of x, y, z, as A', B, C are of x', y', z' subject only to the condition just given.

It would be extremely easy so to modify the preceding theory, as to adapt it to a body whose magnetic particles are regularly arranged,by using the equation (a) in the place of the equation (b) of the preceding article; but, as observation has not yet offered any thing which would indicate a regular arrangement of magnetic particles, in any body hitherto examined, it seems superfluous to introduce this degree of generality, more particularly as the omission may be so easily supplied.

(16.) As an application of the general theory contained in the preceding article, suppose the body A to be a hollow spherical shell of uniform thickness, the radius of whose inner surface is a, and that of its outer one a,; and let the forces inducing a magnetic state in A, arise from any bodies whatever, situate at will, within or without the shell. Then since in the interior of A's mass 0=84, and 0=8V, we shall have (Méc. Cél. Liv. 3)

&=Σ$W +Σ, and VΣU®μ+£U® ̄1;

=

being the distance of the point p, to which & and V belong, from the shell's centre, p, p, &c.,—U“,U", &c. functions of and, the two other polar co-ordinates of p, whose nature has been fully explained by Laplace in the work just cited; the finite integrals extending from i=0 to i= ∞o.

If now, to prevent ambiguity, we enclose the r of equation (6) Art. 15 in a parenthesis, it will become

(r) representing the distance p, do, and the integral extending over both surfaces of the shell. At the inner surface we have

and ra: hence the part of y due to this surface is

do do dr

Σ

- [ &% £ip® a2 + [ do & (i + 1) 6," a**;

the integrals extending over the whole of the inner surface, and do being one of its elements. Effecting the integrations by the formulæ of Laplace (Méc. Céleste, Liv. 3), we immediately obtain the party, due to the inner surface, viz.

Απα

Σ

g

(2i + 1) pï {− ia11μ” + (i + 1) $,“ a+3}.

In the same way the part of due to the outer surface, by observing that for it d__

Απα Σ

аф

dw

=

and r = a,,

is found to be

The sum of these two expressions is the complete value of y, which, together with the values of and V before given, being substituted in the equation (c) Art. 15, we obtain