Which agrees with the equation (b), seeing that

(15.) Conceive now a body A, of any form, to have a magnetic state induced in its particles by the influence of exterior forces, it is clear that if du be an element of its volume, the value of the potential function arising from this element, at any point p' whose co-ordinates are x', y', z', must, since the total quantity of magnetic fluid in dv is equal to zero, be of the form

dv {X (x − x) + Y (y' − y) + Z (e' − e)}

(a);

x,y,, being the co-ordinates of dv, r the distance p', de and X, Y, Z, three quantities dependant on the magnetic state induced in dv, and serving to define this state. If therefore de' be an infinitely small volume within the body A and inclosing the point p', the potential function arising from the whole A exterior to du', will be expressed by

the integral extending over the whole volume of 4 exterior to do'.

It is easy to show from this expression that, in general, although do' be infinitely small, the forces acting in its interior vary in magnitude and direction by passing from one part of it to another; but, when du' is spherical, these forces are sensibly constant in magnitude and direction, and consequently, in this case, the value of the potential function induced in do' by their action, may be immediately deduced from the preceding article.

Let'represent the value of the integral just given, when de' is an infinitely small sphere. The force acting on p' arising from the mass exterior to dv', tending to increase x', will be

the line above the differential coefficient indicating that it is to be obtained by supposing the radius of de' to vanish after differentiation, and this may differ from the one obtained by first making the radius vanish, and afterwards differentiating the resulting function of x', y', z', which last being represented as dif

usual by

dx'

d

we have

= −
[dx dy dɛ X (x' — x) + Y (y' − y) +Z (z'−s),

da da

d

dz

dr = √ dx dy da X (w' - x) + Y ( y − y) + Z (e'− s);

the first integral being taken over the whole volume of A exterior to dv', and the second over the whole of A including dv'. Hence

dy' dy dx' dx

=

d

dx' [dx dyds X (x'− x) + Y ( y −y)+Z (z'− z),

the last integral comprehending the volume of the spherical particle do' only, whose radius a is supposed to vanish after differentiation. In order to effect the integration here indicated, we may remark that X, Y and Z are sensibly constant within dv', and may therefore be replaced by X, Y, and Z,, their values at the centre of the sphere do', whose co-ordinates are x,, y1, z,; the required integral will thus become

fdx dy dz X, (x' − x) + Y, (y' − y) + Z, (e' — ») ̧

Making for a moment E-Xx+Yy+Z, we shall have

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; being the distance p', do, and dw measured from the surface towards the interior of dv'. Now dE

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

da

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

with electricity whose density is da'

and 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 + Y, sin cos + Z, sin ◊ sin ∞);

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

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

[do dE
r da

[do

=

·πr' {X, cos 6+ Y, sin e' cos ' + Z, sin o sin '},

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

[do dE

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

Hence we deduce successively

dy' dy d

dx-da-qz fdx dy dz X (x' — x) + Y (y' − y) + Z (z' — z)

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

But

dy

da expresses the value of the force acting in the

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

dy'
dx'

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'
dx'

and therefore the total force in the direction of a 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 du', 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 dv' by the action of the forces X, Y, Z, is of the form

a' (X cosa + Y cos 8+Zcos y).

being the distance p", dv', 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',

=

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

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

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

since the quantities x", 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'.