Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/97

magnetization of the ellipsoid together with that due to the external magnetic force.

438.] The only case of practical importance is that in which

$\kappa '_{1}=\kappa '_{2}=\kappa '_{3}=0.$

(9)

We have then

${\begin{aligned}A={\frac {\kappa _{1}}{1+\kappa _{1}L}}X,\\B={\frac {\kappa _{2}}{1+\kappa _{2}M}}Y,\\C={\frac {\kappa _{3}}{1+\kappa _{3}N}}Z.\end{aligned}}$

(10)

If the ellipsoid has two axes equal, and is of the planetary or flattened form,

$b=c={\frac {a}{\sqrt {1-e^{2}}}};$

(11)

${\begin{aligned}L&=4\pi \left({\frac {1}{e^{2}}}-{\frac {\sqrt {1-e^{2}}}{e^{3}}}\sin ^{-1}e\right),\\M=N&=2\pi \left({\frac {\sqrt {1-e^{2}}}{e^{3}}}\sin ^{-1}e-{\frac {1-e^{2}}{e^{2}}}\right).\end{aligned}}$

(12)

If the ellipsoid is of the ovary or elongated form

$a=b={\sqrt {1-e^{2}}}c;\,$

(13)

${\begin{aligned}L=M&=2\pi \left({\frac {1}{e^{2}}}-{\frac {1-e^{2}}{2e^{2}}}\log {\frac {1+e}{1-e}}\right),\\N&=4\pi \left({\frac {1}{e^{2}}}-1\right)\left({\frac {1}{2e}}\log {\frac {1+e}{1-e}}-1\right).\end{aligned}}$

(14)

In the case of a sphere, when e = 0,

$L=M=N={\frac {4}{3}}\pi .$

(15)

In the case of a very flattened planetoid L becomes in the limit equal to 4π, and M and N become $\pi ^{2}{\frac {a}{c}}$.

In the case of a very elongated ovoid L and M approximate to the value 2π, while N approximates to the form

$4\pi {\frac {a^{2}}{c^{2}}}\left(\log {\frac {2c}{a}}-1\right)$

and vanishes when e = 1.

It appears from these results that—

(1) When κ, the coefficient of magnetization, is very small, whether positive or negative, the induced magnetization is nearly equal to the magnetizing force multiplied by κ, and is almost independent of the form of the body.