1

dE
de de

which since dE=0, and 8 = 0, reduces itself by what is proved

the integral extending over the whole surface of the sphere dv', of which do is an element; r being the distance p', do, and dw measured from the surface towards the interior of dv'. Now

dE

[do de expresses the value of the potential function for a point

r da

p', within the sphere, supposing its surface everywhere covered

with electricity whose density is

dE

and
"
da

may very easily be obtained by No. 13, Liv. 3, Méc. Céleste. In fact, using for a moment the notation there employed, supposing the origin of the polar co-ordinates at the centre of the sphere, we have

E=E,+ a (X, cos 0+ Y, sin @ cos + Z, sin sin );

E, being the value of E at the centre of the sphere. Hence

dE
da

=

X, cos + Y, sin cos + Z, sin @ sin ☎,

and as this is of the form U(Vide Méc. Céleste, Liv. 3), we immediately obtain

TX, cos + Y, sin e' cos '+Z, sine sin '},

where r', ',' are the polar co-ordinates of p'. Or by restoring x', y' and z'

fdo dE

jdo

x + = {π {X, (x − x,) + Y, (y' − y) + Z, (z' — z,)}.
r da

Hence we deduce successively

d

− −
— z)

dy_dy - 2 dx dy dz X (x' - a) + Y (y' − y) + Z (a' − 2)

dx

-

da

If now we make the radius a vanish, X, must become equal to X', the value of X at the point p', and there will result

dy

But

dx'

expresses the value of the force acting in the

direction of a positive, on a point p' within the infinitely small sphere de', arising from the whole of A exterior to dv; substituting now for force becomes

its value just found, the expression of this

Supposing V' to represent the value of the potential function at p', arising from the exterior bodies which induce the magnetic state of A, the force due to them acting in the same direction, is

dV'
dac'

and therefore the total force in the direction of x' positive, tending to induce a magnetic state in the spherical element dv', is

In the same way, the total forces in the directions of y' and ' positive, acting upon dv', are shown to be

By the equation (b) of the preceding article, we see that when de' is a perfect conductor of magnetism, and its particles are not regularly arranged, the value of the potential function at any point p", arising from the magnetic state induced in do' by the action of the forces X, Y, Z, is of the form

being the distance p", du', and a, B, y the angles which r' forms with the axes of the rectangular co-ordinates. If then x", y", z" be the co-ordinates of p", this becomes, by observing that here a = kdv',

kdv' {X (x" — x') + Y (y" − y) + Z' (z′′"—z')}

k being a constant quantity dependant on the nature of the body. The same potential function will evidently be obtained from the expression (a) of this article, by changing dv, p', and their co-ordinates, into dv', p", and their co-ordinates; thus we

have

dv′ {X' (x" — ») + Y' (y" — y') + Z (z′′ — z')} ̧

13

Equating these two forms of the same quantity, there results the three following equations:

since the quantities a", y', z′′ are perfectly arbitrary. Multiplying the first of these equations by dx', the second by dy', the third by dz', and taking their sum, we obtain

0 = (1 — §πk) (X'dx' + Y'dy' + Z'dz') + k'dy' + kd V'.

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

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

the above, by integration, becomes

const. = (1 − fπk) p' + 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 du', 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

seeing that

3dv X(x" -x)+ Y (y" — y') + Z (z" — z')

4π

3dv
4π

= a; 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, or, which is the same thing,

3g k 4π

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

(x

✦' — [dz dy de X (2' — 2) + Y (y' − y) + Z (e'− 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;

the quantities

d

and belonging to this element. We have, dw

and consequently 8p=0; the symbol & referring to x', y', z' the co-ordinates of p'; or, since x', y' and ' are arbitrary, by making them equal to x, y, z respectively, there results

0 = 0,

in virtue of which, the value of y', by Article 3, becomes

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

former equation serving to determine p' gives, by changing