Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/110

Equating these values, and remembering that β = α - θ, we find

$\tan \theta ={\frac {D\sin \alpha }{X+D\cos \alpha }}$

(1)

to determine the direction of the axis after deflexion.

We have next to find the intensity of magnetization produced in the mass by the force X, and for this purpose we must resolve the magnetic moment of every molecule in the direction of x, and add all these resolved parts.

The resolved part of the moment of a molecule in the direction of x is

$m\cos \theta .\,$

The number of molecules whose original inclinations lay between α and α + dα is

${\frac {n}{2}}\sin \alpha \,d\alpha .$

We have therefore to integrate

$I=\int _{0}^{\pi }{{\frac {mn}{2}}\cos \theta \sin \alpha \,d\alpha ,}$

(2)

remembering that θ is a function of α.

We may express both θ and α in terms of R, and the expression to be integrated becomes

${\frac {mn}{4X^{2}D}}(R^{2}+X^{2}-D^{2})dR,$

(3)

the general integral of which is

${\frac {mnR}{12X^{2}D}}(R^{2}+3X^{2}-D^{2})+C.$

(4)

In the first case, that in which X is less than D, the limits of integration are R = D + X and R = D – X. In the second case, in which X is greater than D, the limits are R = X + D and R = X – D.

When X is less than D,

$I={\frac {2}{3}}{\frac {mn}{D}}X.$

(5)

When X is equal to D,

$I={\frac {2}{3}}mn.$

(6)

When X is greater than D,

$I=mn\left(1-{\frac {1}{3}}{\frac {D^{2}}{X^{2}}}\right);$

(7)

and when X becomes infinite

$I=mn.\,$

(7)

According to this form of the theory, which is that adopted by Weber^[1], as the magnetizing force increases from 0 to D, the

1. There is some mistake in the formula given by Weber (Trans. Acad. Sax. i. p. 572 (1852), or Pogg., Ann. Lxxxvii. p. 167 (1852)) as the result of this integration, the steps of which are not given by him. His formula is

   $I=mn{\frac {X}{\sqrt {x^{2}+D^{2}}}}{\frac {x^{4}+{\frac {7}{6}}X^{2}D^{2}+{\frac {2}{3}}D^{4}}{X^{4}+X^{2}D^{2}+D^{4}}}.$