and consequently

V+DV=B (x cos + y sin w) +hDp (y cos x sin )

cos ≈ cos (Þ — ) Do},

cos ($ − ∞) D4} .

is augmented by the infinitely small angle

b

Do, ≈ receives the corresponding increment-cos (4 – ) D4,

and the form of V remains unaltered; the preceding reasoning is consequently applicable to every instant, and the general relation between and expressed by

H being an arbitrary constant, and y, as in the former part of this article, the smallest root of

0=b sin y-B.

Let, and o be the initial values of and ; then the total potential function at the next instant, if the electric fluid remained fixed, would be

V=B (x cos + y sin ) + (x sin 4. — y cos 4) bdp,

and the whole force to move a particle p, whose co-ordinates

0

which, in order that our solution may be applicable, must not be less than ẞ, and consequently the angle - must be between 0 and 7: when this is the case, is immediately determined from by what has preceded. In fact, by finding the value of H from the initial values, and 4, and making – 10, we obtain

tan

tan = e(Þ¬Þ0) coty + tan y tan (e-) coty

being the initial value of ¿.

We have, in the latter part of this article, considered the body A at rest, and the line X', parallel to the direction of b, as revolving round it: but if, as in the former, we now suppose this line immovable and the body to turn the contrary way, so that the relative motion of X' to X may remain unaltered, the electric state of the body referred to the axes X, Y, Z, evidently depending on this relative motion only, will consequently remain the same as before. In order to determine it on the supposition just made, let X' be the axis of x', one of the co-ordinates of p, referred to the rectangular axes X', Y', Z, also y', z, the other two; the direction X' Y', being that in which A revolves. Then, if 'be the angle the system of lines L, L', &c. forms with the plane (x', z), we shall have

@ + w' = 0;

, as before stated, being the angle included by the axes X, X'. Moreover the general values of V and will be

V=B (x' cos a' + y' sin a') and = 1π + 1y - 1∞',

and the initial condition, in order that our solution may be applicable, will evidently become -☎=' = a quantity betwixt 0 and π.

As an example, let tan y =

=

0

since we know by experiment

that y is generally very small; then taking the most unfavourable case, viz. where ', 0, and supposing the body to make one revolution only, the value of , determined from its initial one, +-', will be found extremely small and only

=

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

Hence, the polarity induced by the rotation is ultimately directed along a line, making an angle equal to +y 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 fluid 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 to arise

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)

12

U (0) U (1) U (2)

+ &c.;

+ 3

r' = √ (x' + y22 + z") being the distance Op'.

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

U (0)

12

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 contain x, y, z.

may

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

313

a", b", c", e", f", 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