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Which agrees with the equation (6), seeing that

005 a = , cos B =, cosy

(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 do is equal to zero, be of the form dv {X (c' – ) + Y (3y) + 2(2— 2)}

... (a);

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

Idady dz ? (x– 2} + Y(y y) +Z(e' — 2);

the integral extending over the whole volume of A 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 dv' by their action, may be immediately deduced from the preceding article.

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

the line above the differential coefficient indicating that it is to be obtained by supposing the radius of do' 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 ac', y', z, which last being represented as usual by any , we have

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

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the last integral comprehending the volume of the spherical particle du' 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 2, their values at the centre of the sphere dv', whose co-ordinates are 2,, Y., 2,; the required integral will thus become

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

expresses the value of the potential function for a point p', within the sphere, supposing its surface everywhere covered with electricity whose density is , 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 0 + Y sin 8 cos w + Z, sin 0 sina);

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

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where r', 0, a' are the polar co-ordinates of p. Or by restoring ac', y' and '

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Hence we deduce successively des de les do dy de 1 (+ = ) + Yly = y) +Z(z'—-) - Mano duomen 10 {X, (d* – 2) + 7, 69-)

+2,(z'- 2,)} = 17X. 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

they are pe = forX', i.e. als de -fox”. But Love expresses the value of the force acting in the direction of x positive, on a point p within the infinitely small sphere du', arising from the whole of A exterior to do'; substituting now for its value just found, the expression of this force becomes

4*X' - ab



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

d '
dixi :

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

wy, dh dy'

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

for" - autem

in - Y, and, In2" - det er =

By the equation (6) of the preceding article, we see that when du' 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

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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, é', and their co-ordinates, into du, p", and their co-ordinates ; thus we have

đó X' (z” - a) + Y (i3) +Z(=" – 4)}

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

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since the quantities a", y", 2" are perfectly arbitrary. Multiplying the first of these equations by da', the second by dy', the third by dz', and taking their sum, we obtain

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