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')2 + a)"} we shall have

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

very nearly; the two limits of the integral being - μ and + μ and 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 -λ to x +λ.

x=

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 a❜AX;

dX""

making now x=+λ, zg a is shown to be of the order

dx

dX""

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

dx

order aA; but for any other value of x the function multiplying dx" becomes of the order a, and therefore we may without da sensible error neglect the term containing it, and likewise suppose

dX"
dx

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 dx, 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 ƒ, 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

-X

A' (μ'-≈ -- μ'+x) dx.

This expression agrees precisely with the one before deduced

from theory, and gives, for the determination of the constants A' and ', the equations

The chapter in which these experiments are related, contains also a number of results, relative to the forces with which magnetized wires tend to turn towards the meridian, when retained at a given angle from it, and it is easy to prove that this force for a fine wire, whose variable section is s, will be proportional to the quantity

аф
dx'

where the wire is magnetized in any way either to saturation or otherwise, the integral extending over its whole length. But in a cylindric wire magnetized to saturation, we have, by neglecting quantities of the order a1,

and therefore for this wire the force in question is proportional to

The value of g, dependent on the nature of the substance of which the needles are formed, being supposed given as it ought to be, we have only to determine 8 in order to compare this result with observation. But ẞ depends upon A = 2 log, and

α

on account of the smallness of a, A undergoes but little alteration for very considerable variations in μ, so that we shall be able in every case to judge with sufficient accuracy what value of μ ought to be employed: nevertheless, as it is always desirable to avoid every thing at all vague, it will be better to determine A by the condition, that the sum of the squares of the errors d'X' committed by employing, as we have done, A for the apdx'

the whole length of the wire. In this way I find when A is so

1

great that quantities of the order may be neglected,

βλ

A.231863-2 log aẞ+2aẞ;

where .231863 &c. = 2 log 2 − 2 (A); (A) being the quantity represented by A in LACROIX' Traité du Cal. Diff. Tome 3, p. 521. Substituting the value of A just found in the equation

We hence see that when the nature of the substance of which the wires are formed remains unchanged, the quantity aß is constant, and therefore ẞ varies in the inverse ratio of a. This agrees with what M. BIOT has found by experiment in the chapter before cited, as will be evident by recollecting that

B=-log μ'.

From an exp riment made with extreme care by COULOMB,

1

on a magnetized wire whose radius was inch, M. BIOT has

12

found the value of μ' to be .517948 (Traité de Phy. Tome 3, p. 78). Hence we have in this case

which, according to a remark just made, ought to serve for all steel wires. Substituting this value in the equation (a) of the present article, we obtain

g=.986636.

With this value of g we may calculate the forces with which

different lengths of a steel wire whose radius is

1 12

inch, tend to