Equating the coefficients of like powers of the variable r, we have generally, whatever i may be,

0 = (1-g)," +

0 = (1 − g) $(+

3g

3g

(2i+1) a,† {ia, ̃4”—(i+1)6,"a‚13}+&% U®;

neglecting the constant on the right side of the equation in r as superfluous, since it may always be made to enter into 0). If now, for abridgment, we make

D = (2i + 1)* (1 + g) + (i − 1) (i + 2) go — 9g'i (i + 1)

we shall obtain by elimination

3g(i+1)(2i+1)a,

{2i+1)(2i+1+(i+2)g} _ 39 u ̧μ3g(i+1) (26

3g Two __

=

4π

D

24+1

Uw3gi a' 3g

4π

3gi (2i+1) a¤1 __ 39 U ̧μ) (2i+1) (2i+1+(i'−1) g} ̧

D

These values substituted in the expression

D

give the general value of ø in a series of the powers of r, when the potential function due to the bodies inducing a magnetic state in the shell is known, and thence we may determine the value of the potential function arising from the shell itself, for any point whatever, either within or without it.

When all the bodies are situate in the space exterior to the shell, we may obtain the total actions exerted on a magnetic particle in its exterior, by the following simple method, applicable to hollow shells of any shape and thickness.

The equation (c) Art. 15 becomes, by neglecting the superfluous constant,

If now (4) represent the value of the potential function, corresponding to the value of p at the inner surface of the shell, each of the functions (p), and V, will satisfy the equations

0 = 8(4), 0 = d¥ and 0=8V, and moreover, have no singular values in the space within the shell; the same may therefore be said of the function

and as this function is equal to zero at the inner surface, it follows (Art. 5) that it is so for any point p of the interior space. Hence

But

+V is the value of the total potential function at the point p, arising from the exterior bodies and shell itself: this function will therefore be expressed by

In precisely the same way, the value of the total potential function at any point p', exterior to the shell, when the inducing bodies are all within it, is shown to be

(p) being the potential function corresponding to the value of at the exterior surface of the shell. Having thus the total potential functions, the total action exerted on a magnetic particle in any direction, is immediately given by differentiation.

To apply this general solution to our spherical shell, the inducing bodies being all exterior to it, we must first determine p, the value of at its inner surface, making 0 = EU since there are no interior bodies, and thence deduce the value of (p). Substituting for 4" and " their values before given, making U=0 and ra, we obtain

and the corresponding value of (4) is (Méc. Cél. Liv. 3)

The value of the total potential function at any point p within the shell, whose polar co-ordinates are r, 0, w, is

47

-3y

3g

(1 − g) (6) = (1 − g) (1 + 2g) ΣU« (2i+1)* ‚ ̧

D

In a similar way, the value of the same function at a point p' exterior to the shell, all the inducing bodies being within it, is found to be

r, and w in this expression representing the polar co-ordinates of p'.

To give a very simple example of the use of the first of these formulæ, suppose it were required to determine the total action exerted in the interior of a hollow spherical shell, by the magnetic influence of the earth; then making the axis of x to coincide with the direction of the dipping needle, and designating by f, the constant force tending to impel a particle of positive fluid in the direction of x positive, the potential function V, due to the exterior bodies, will here become

V=-f.x-fcos 0.r=U.r.

The finite integrals expressing the value of V reduce themselves therefore, in this case, to a single term, in which i=1, and the corresponding value of D being

We therefore see that the effect produced by the intervening shell, is to reduce the directive force which would act on a very

small magnetic needle,

from ƒ, to

1+g-2g1

af.

1+7-2g3 a

In iron and other similar bodies, g is very nearly equal to 1, and therefore the directive force in the interior of a hollow spherical shell is greatly diminished, except when its thickness is very small compared with its radius, in which case, as is evident from the formula, it approaches towards the original value ƒ, and becomes equal to it when this thickness is infinitely small.

To give an example of the use of the second formula, let it be proposed to determine the total action upon a point p, situate on one side of an infinitely extended plate of uniform thickness, when another point P, containing a unit of positive fluid, is placed on the other side of the same plate considering it as a perfect conductor of magnetism. For this, let fall the perpendicular PQ upon the side of the plate next P, on PQ prolonged, demit the perpendicular pq, and make PQ-b, Pq=u, pq=v, and the thickness of the plate; then, since its action is evidently equal to that of an infinite sphere of the same thickness, whose centre is upon the line QP at an infinite distance from P, we shall have the required value of the total potential function at by supposing a, = a+t, a infinite, and the line PQ prolonged to be the axis from which the angle is measured. Now in the present case

and the value of the potential function, as before determined, is

(1 − 9) (1 − 29) Σ (2i + 1)o Um+.

-

D

From the first expression we see that the general term is a quantity of the order (a - b)'r. Moreover, by substituting for r its value in u.

(a - b)' +1 = (a — b)* (a − b + u) → 1 = 1 e ̄ = ;

1

ww.looting such quantities as are of the order compared with

a

hp retained. The general term U, and consequently

In the finite integrals just given, the increment of i is 1, and

the corresponding increment of y is

1

a

=dy (because a is in

finite), the finite integrals thus change themselves into ordinary In fact (Méc. Cél. Liv. 3), U“ always

integrals or fluents.

satisfies the equation

and as is infinitely small whenever V has a sensible value, we may eliminate it from the above by means of the equation av, and we obtain by neglecting infinitesimals of higher

orders than those retained, since

T

+ y2U" = 0.

seeing that the remaining part of the general integral becomes infinite when v vanishes, and ought therefore to be rejected. It now only remains to determine the value of the arbitrary constant A. Making, for this purpose, 0, i. e. v = 0, we

have

U!" = (a - b)' and [ d = 1π : hence («—b)' = }(4π),

。 √ (1 − B2)

By substituting for A and r their values, there results