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equal to a unit in the 27th decimal place. We thus see with what rapidity & decreases, and consequently, the body approaches to a permanent state, defined by the equation
0= $= ft + ty-fo. Hence, the polarity induced by the rotation is ultimately directed along a line, making an angle equal to fo try with the axis X', which agrees with what was shown in the former part of this article.
The value of V at the body's surface being thus known at any instant whatever, that of the potential function at a point p exterior to the body, together with the forces acting there, will be immediately determined as before.
APPLICATION OF THE PRELIMINARY RESULTS
TO THE THEORY OF MAGNETISM.
(14.) The electric fluid appears to pass freely from one part of a continuous conductor to another, but this is by no means the case with the magnetic fluid, even with respect to those bodies which, from their instantly returning to a natural state the moment the forces inducing a magnetic one are removed, must be considered, in a certain sense, as perfect conductors of magnetism. Coulomb, I believe, was the first who proposed to consider these as formed of an infinite number of particles, each of which conducts the magnetic Auid in its interior with perfect freedom, but which are so constituted that it is impossible there shall be any communication of it from one particle to the next. This hypothesis is now generally adopted by philosophers, and its consequences, as far as they have hitherto been developed, are found to agree with observation; we will therefore admit it in what follows, and endeavour thence to deduce, mathematically, the laws of the distribution of magnetism in bodies of any shape whatever.
Firstly, let us endeavour to determine the value of the potential function, arising from the magnetic state induced in a very small body A, by the action of constant forces directed: parallel to a given right line; the body being composed of an infinite number of particles, all perfect conductors of magnetism and originally in a natural state. In order to deduce this more immediately from Art. 6, we will conceive these forces tu arise
from an infinite quantity of magnetic fluid, concentrated in a point p on this line, at an infinite distance from A. Then the origin O of the rectangular co-ordinates being anywhere within A, if x, y, a, 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)
U (0) 0 (1) 003
V= + tumas + &c.; g' = v(a' +ya + &') being the distance Op.
Moreover, since the total quantity of magnetic Auid in A is equal to zero, U = 0. Supposing now gı' very great compared 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 CM its most general value, we obtain
A, B, C, being quantities independent of a', y', a', but which may contain 2, y, ..
Now (Art. 6) the value of 7 will remain unaltered, when we change x, y, z, into we', y', s', and reciprocally. Therefore
A, B', C", being the same functions of x', 4, 5, as A, B, C, are of x, y, z. Hence it is easy to see that V must be of the form
If X, Y, Z, represent the forces arising from the magnetism concentrated in p, in the directions of x, y, e, positive, we shall have
a', 6, &c. being other constant quantities. But it will always be possible to determine the situation of three rectangular axes, 80 that e, f, and g may each be equal to zero, and consequently V be reduced to the following simple form v_aXx' +6 Yy' + cZz
a, b, and c being three constant quantities.
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', 6', 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'='=', é'= 0, f'= 0, and g'= 0, therefore in this case
a' (Xic' + Yy + Z2') V=
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 V reduce itself to
y = d' Xx' + B' (Yy' + Z:')
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 saeans 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 particles being arranged in a confused manner, will, in fact, form the basis of our theory, and althougli 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 tho point p', arising from A, must evidently be of the form
u representing as before the distance Op', and 0 the angle formed botween the line Op', and another line OD fixed in A. If now f be the magnitude of the force directed along OB, the constant la will evidently be of the form k= a'f; a' being a constant quantity. The value of V, just given, holds good for any arrangenent, regular or irregular, of the magnetic particles composing A, 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 0 - angle BOp' and
Let now a, B, y, be the angles that the line Op' = p makes with the axes of x, y, , and a', B, q', those which OB makes with the same axes; then, substituting for cos 6 its value
cos a cos a' + cos ß cos B'+ cos y cosy', we have, since
f cosa = X, f cos B = Y, fcosy= 2,