Page:ThomsonMagnetic1889.djvu/7

Equating the coefficients of $\epsilon ^{ipt}$ to zero in equations (2) and (3) we have

$-{\dfrac {\sigma }{4\pi \mu '}}\nabla ^{2}\mathrm {F} _{1}=-{\dfrac {d\mathrm {F} _{1}}{dt}}-{\dfrac {d\psi _{1}}{dx}}$ inside the sphere,

with similar equations for $\mathrm {G} _{1}$ and $\mathrm {H} _{1}$; outside the sphere we have

$-{\dfrac {1}{\mu \mathrm {K} }}\nabla ^{2}\mathrm {F} _{1}=-{\dfrac {\partial ^{2}\mathrm {F} _{1}}{\partial t^{2}}}-{\dfrac {d^{2}\psi _{1}}{dt\ dx}}+{\dfrac {\omega e}{\mathrm {K} }}{\dfrac {d^{2}}{dx\ dz}}{\dfrac {1}{r}},$

(4)

with similar equations for $\mathrm {G} _{1}$ and $\mathrm {H} _{1}$.

The form of equation (4) suggests that we should put

$\psi _{1}=B{\dfrac {d}{dz}}{\dfrac {1}{r}}$

A particular integral of (4) is then

$\mathrm {F} _{1}={\dfrac {c\mathrm {B} }{p}}{\dfrac {d^{2}}{dx\ dz}}{\dfrac {1}{r}}-{\dfrac {\omega e}{\mathrm {K} p^{2}}}{\dfrac {d^{2}}{dx\ dz}}{\dfrac {1}{r}}.$

The complementary function is that solution of the differential equation

$\nabla ^{2}\mathrm {F} _{1}+\mu \mathrm {K} p^{2}\mathrm {F} _{1}=0$

which, when considered as a function of the angular coordinates of a point, varies as $r^{3}{\dfrac {d^{2}}{dx\ dz}}{\dfrac {1}{r}}$; this (see Proc. Math. Soc. vol. xv. p. 212) is

$\mathrm {C} \mathrm {E} _{2}(i\lambda r)r^{3}{\dfrac {d^{2}}{dx\ dz}}{\dfrac {1}{r}}$

where

$\mathrm {E} _{2}(i\lambda r)={\dfrac {3\epsilon ^{-i\lambda r}}{(i\lambda r)^{3}}}+{\dfrac {3\epsilon ^{-i\lambda r}}{(i\lambda r)^{2}}}+{\dfrac {\epsilon ^{-i\lambda r}}{i\lambda r}}$

and

$\lambda ^{2}=\mu \mathrm {K} p^{2}$

Thus, outside the sphere,

${\begin{array}{l}\mathrm {F} _{1}=\mathrm {C} \mathrm {E} _{2}(i\lambda r)r^{3}{\dfrac {d^{2}}{dx\ dz}}{\dfrac {1}{r}}+{\dfrac {i\mathrm {B} \epsilon ^{pt}}{p}}{\dfrac {d^{2}}{dx\ dz}}{\dfrac {1}{r}}-{\dfrac {\omega e\epsilon ^{pt}}{\mathrm {K} p^{2}}}{\dfrac {d^{2}}{dx\ dz}}{\dfrac {1}{r}},\\\\\mathrm {G} _{1}=\mathrm {C} \mathrm {E} _{2}(i\lambda r)r^{3}{\dfrac {d^{2}}{dy\ dz}}{\dfrac {1}{r}}+{\dfrac {i\mathrm {B} \epsilon ^{pt}}{p}}{\dfrac {d^{2}}{dy\ dz}}{\dfrac {1}{r}}-{\dfrac {\omega e\epsilon ^{pt}}{\mathrm {K} p^{2}}}{\dfrac {d^{2}}{dy\ dz}}{\dfrac {1}{r}},\\\\\mathrm {H} _{1}=\mathrm {C} \mathrm {E} _{2}(i\lambda r)r^{3}{\dfrac {d^{2}}{dz^{2}}}{\dfrac {1}{r}}-2\mathrm {C} \mathrm {E} _{2}(i\lambda r)+{\dfrac {i\mathrm {B} }{p}}\epsilon ^{pt}{\dfrac {d^{2}}{dz^{2}}}{\dfrac {1}{r}}-{\dfrac {\omega e\epsilon ^{pt}}{\mathrm {K} p^{2}}}{\dfrac {d^{2}}{dz^{2}}}{\dfrac {1}{r}},\end{array}}$

where $\mathrm {E} _{0}(i\lambda r)=\epsilon ^{-i\lambda r}/i\lambda r$, and is introduced into the expression