will, in the first place, suppose a cylindric wire whose radius is a and length 2, is exposed to the action of a constant force, equal to f, and directed parallel to the axis of the wire, and then endeavour to determine the magnetic state which will thus be induced in it. For this, let r be a perpendicular falling from a point p within the wire upon its axis, and x, the distance of the foot of this perpendicular from the middle of the axis; then f being directed along x positive, we shall have for the value of the potential function due to the exterior forces

V=-fx,

and the equations (b), (c) (art. 15) become, by omitting the superfluous constant,

(r), the distance p', do being inclosed in a parenthesis to prevent ambiguity, and p' being the point to which belongs. By the same article we have 0=84 and 0= &, and as and evidently depend on x and r only, these equations being written at length are

Sincer is always very small compared with the length of the wire, we may expand 4 in an ascending series of the powers of r, and thus

$=X+Xr+X‚ ̧μ3 + etc.;

X, X, X, etc. being functions of x only. By substituting this value in the equation just given, and comparing the coefficients of like powers r, we obtain

In precisely the same way the value of y is found to be

It now only remains to find the values of X and Y in functions of x. By supposing p' placed on the axis of the wire, the equation (c) becomes

Y-- [de (d);

Y=

the integral being extended over the whole surface of the wire: Y' belonging to the point p', whose co-ordinates will be marked with an accent.

The part of Y' due to the circular plane at the end of the cylinder, where x=-λ, is

of the order a on account of their smallness; X" representing

the value of X when x=- -λ.

At the other end where x =+λ we have do = 2πrdr,

dx" 2πrdr

(r)

dX"
dx

=2π · [√ { (λ − x')2 + a3} −λ+x'];

X" designating the value of X when x=+λ.

At the curve surface of the cylinder

do=2madx and

аф аф du

dr

provided we omit quantities of the order a compared with those retained. Hence the remaining part due to this surface is

the integral being taken from x =−λ to x=+λ. The total value of Y' is therefore

the limits of the integral being the same as before. If now we substitute for (r) its value √(x-x') + a)"} we shall have A

- πα

dx d'X (2) dx*

== πα

'{(x − x′)2+ a3} dx3 i

both integrals extending from x = − λ to x=+λ.

On account of the smallness of a, the elements of the last integral where x is nearly equal to x' are very great compared with the others, and therefore the approximate value of the expression just given, will be

-

a

very nearly; the two limits of the integral being – μ and +μ and μ so chosen that when p' is situate anywhere on the wire's axis, except in the immediate vicinity of either end, the approximate shall differ very little from the true value, which may in every case be done without difficulty. Having thus, by substitution, a value of Y' free from the sign of integration, the value of Y is given by merely changing x' into x and X' into X; in this way

The equation (c), by making r = 0, becomes

an equation which ought to hold good, for every value of x, from x=-λ to x=+λ.

In those cases to which our theory will be applied, 1-g is a small quantity of the same order as a A, and thus the three terms of the first line of our equation will be of the order a1AX;

dx" making now x=+λ, } 9 = dx

dx"

a is shown to be of the order

a'AX", and therefore ÷ X" is a small quantity of the

dx

order a4; but for any other value of x the function multiplying dX"

becomes of the order a", and therefore we may without dx sensible error neglect the term containing it, and likewise

In the same way by making x=-λ, it may be shown that the

But the density of the fluid at the surface of the wire, which would produce the same effect as the magnetized wire itself, is

and therefore the total quantity in an infinitely thin section whose breadth is do, will be

As the constant quantity ƒ may represent the coercive force of steel or other similar matter, provided we are allowed to suppose this force the same for every particle of the mass, it is clear that when a wire is magnetized to saturation, the effort it makes to return to a natural state must, in every part, be just equal to f; and therefore, on account of its elongated form, the degree of magnetism retained by it will be equal to that which would be induced in a conducting wire of the same form by the force f, directed along lines parallel to its axis. Hence the preceding formulæ are applicable to magnetized steel wires. But it has been shown by M. BIOT (Traité de Phy. Tome 3, Chap. 6) from COULOMB's experiments, that the apparent quantity of free fluid in any infinitely thin section is represented by

A' (u'-x — μ'+x) dx.

This expression agrees precisely with the one before deduced