from an infinite quantity Q of magnetic fluid, concentrated in a point p on this line, at an infinite distance from A. Then the origin of the rectangular co-ordinates being anywhere within A, if x, y, z, be those of the point p, and x', y', z', those of any other exterior point p', to which the potential function V arising from A belongs, we shall have (vide Méc. Cél. Liv. 3)

Moreover, since the total quantity of magnetic fluid in A is equal to zero, U=0. Supposing now very great compared U with the dimensions of the body, all the terms after in the

expression just given will be exceedingly small compared with this, by neglecting them, therefore, and substituting for Uits most general value, we obtain

A, B, C, being quantities independent of x', y', z', but which may contain x, y, z.

Now (Art. 6) the value of V will remain unaltered, when we change x, y, z, into a', y', x', and reciprocally. Therefore

A', B', C', being the same functions of x', y', z', as A, B, C, are of x, y, z. Hence it is easy to see that V must be of the form

V=

a"xx'+b"yy'+c"zz'+e" (xy'+yx') +ƒ"(xz'+zx')+g′′(yz'+zy′)

818

a", b", c", e", ƒ", g", being constant quantities.

If X, Y, Z, represent the forces arising from the magnetism concentrated in p, in the directions of x, y, z, positive, we shall

and therefore Vis of the form

a'Xx'+b'Yy'+c'Zz'+e'(Xy'+Yx')+ƒ'(Xz'+Zx')+q′ (Yz'+Zy')

18

a', b', &c. being other constant quantities. But it will always be possible to determine the situation of three rectangular axes, so that e, f, and g may each be equal to zero, and consequently ✓ be reduced to the following simple form

When A is a sphere, and its magnetic particles are either spherical, or, like the integrant particles of non-crystallized bodies, arranged in a confused manner; it is evident the constant quantities a', b', c', &c. in the general value of V, must be the same for every system of rectangular co-ordinates, and consequently we must have a'=b'= c', e' = o, ƒ'= o, and g'=0, therefore in this case

V _

a' (Xx' + Yy' + Zz')

3

........

(b);

a' being a constant quantity dependant on the magnitude and nature of A.

The formula (a) will give the value of the forces acting on any point p', arising from a mass A of soft iron or other similar matter, whose magnetic state is induced by the influence of the earth's action; supposing the distance Ap' to be great compared with the dimensions of A, and if it be a solid of revolution, one of the rectangular axes, say X, must coincide with the axis of revolution, and the value of reduce itself to

a' and b' being two constant quantities dependant on the form and nature of the body. Moreover the forces acting in the directions of x, y', z', positive, are expressed by

We have thus the means of comparing theory with experiment, but these are details into which our limits will not permit us to

enter.

The formula (b), which is strictly correct for an infinitely small sphere, on the supposition of its magnetic particies being arranged in a confused manner, will, in fact, form the basis of our theory, and although the preceding analysis seems sufficiently general and rigorous, it may not be amiss to give a simpler proof of this particular case. Let, therefore, the origin O of the rectangular co-ordinates be placed at the centre of the infinitely small sphere A, and OB be the direction of the parallel forces acting upon it; then, since the total quantity of magnetic fluid in A is equal to zero, the value of the potential function V, at the point p', arising from A, must evidently be of the form k cos e V= 12 ;

representing as before the distance Op', and 8 the angle formed between the line Op', and another line OD fixed in A. If now f be the magnitude of the force directed along OB, the constant k will evidently be of the form kaf; a' being a constant quantity. The value of V, just given, holds good for any arrangement, regular or irregular, of the magnetic particles composing 4, but on the latter supposition, the value of V would evidently remain unchanged, provided the sphere, and consequently the line OD, revolved round OB as an axis, which could not be the case unless OB and OD coincided. Hence = angle BOp' and

a'f cos 0

12

Let now a, ß, y, be the angles that the line Op'r' makes with the axes of x, y, z, and a', B', y', those which OB makes with the same axes; then, substituting for cos 0 its value

cos a cos a' + cos B cos B' + cos y cos y',

(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 eleruent, 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

x, y, z, being the co-ordinates of dv, r the distance p', dv 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 de', will be expressed by

fdxdy dz X (x' − x) + Y (y' − y) + Z (z' — 2) ;

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

It is easy to show from this expression that, in general, although dv' 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 dv' is an infinitely small sphere. The force acting on p' arising from the mass exterior to dv', tending to increase x', will be

'dy'
dx')

the line above the differential coefficient indicating that it is to be obtained by supposing the radius of dv' 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 dy' we have

usual by

da

dy = d
dx dxdy

dy' d

[dx dy dz X (x'− x) + Y (y'− y) +Z (z'−3),

dx = → [dx dy dz X (x − x) + Y (y'− y) + Z (z' — «);

da

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 = dr2 [ dx dy dø X (x' — x) + Y (y −3)+Z (z'— z); dx Jdx dz

dx'

dx

the last integral comprehending the volume of the spherical particle dv' 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 du', whose co-ordinates are x,, Y1, z,; the required integral will thus become

[ dx dy dz X, (x − x) + Y, (y'− y) +Z, (e'—x) ̧

Making for a moment E=Xx+Yy+Zz, we shall have

x=dE, Y=dE, z=dE

dy

Z