Page:ThomsonMagnetic1889.djvu/8

8 for $$\mathrm{H}$$ to make

$\dfrac{d\mathrm{F}}{dx}+\dfrac{d\mathrm{G}}{dy}+\dfrac{d\mathrm{H}}{dz}=0.$|undefined

Inside the sphere the differential equations for $$\mathrm{F}_{1}$$ and $$\mathrm{G}_{1}$$ are of the form

$-\dfrac{\sigma}{4\pi\mu'}\nabla^{2}\mathrm{F}_{1}=-ip\mathrm{F};$

if $$\lambda_{1}^{2}=-4\pi\mu'ip/\sigma$$, the solution of this equation is

$\mathrm{F}_{1}=\mathrm{D}\epsilon^{ipt}\mathrm{S}_{2}\left(\lambda_{1}r\right)r^{3}\dfrac{d^{2}}{dx\ dz}\dfrac{1}{r},$|undefined

where

$\mathrm{S}_{2}\left(\lambda_{1}r\right)=\dfrac{3\sin\lambda_{1}r}{\left(\lambda_{1}r\right)^{3}}-\dfrac{3\cos\lambda_{1}r}{\left(\lambda_{1}r\right)^{2}}-\dfrac{\sin\lambda_{1}r}{\lambda_{1}r}.$|undefined

Similarly

$\mathrm{G}_{1}=\mathrm{D}\epsilon^{ipt}\mathrm{S}_{2}\left(\lambda r\right)r^{3}\dfrac{d^{2}}{dx\ dz}\dfrac{1}{r};$|undefined

the differential equation for $$\mathrm{H}_{1}$$ is

$-\dfrac{\sigma}{4\mu'}\nabla^{2}\mathrm{H}_{1}=-ip\mathrm{H}_{1}+\dfrac{\mathrm{B}}{a^{3}}.$|undefined

So that

$\mathrm{H}_{1}=\mathrm{D}\epsilon^{ipt}\mathrm{S}_{2}\left(\lambda_{1}r\right)r^{3}\dfrac{d^{2}}{dz^{2}}\dfrac{1}{r}+2\mathrm{D}\epsilon^{ipt}\mathrm{S}_{0}\left(\lambda_{1}r\right)+\dfrac{\mathrm{B}}{ipa^{3}}\epsilon^{ipt},$|undefined

where

$\mathrm{S}_{0}\left(\lambda_{1}r\right)=\dfrac{\sin\lambda_{1}r}{\lambda_{1}r},$

and is introduced to make

$\dfrac{d\mathrm{F}}{dx}+\dfrac{d\mathrm{G}}{dy}+\dfrac{d\mathrm{H}}{dz}=0.$|undefined

Since $$\mathrm{F}_{1},\mathrm{G}_{1},\mathrm{H}_{1}$$ are continuous when $$r=a$$, if $$a$$ is the radius of the sphere we have

{{MathForm2|(5)|$$\left.\begin{array}{c} \mathrm{C}\mathrm{E}_{2}(i\lambda a)-\mathrm{DS}_{2}\left(\lambda_{1}a\right)+\dfrac{i\mathrm{B}}{pa^{3}}=\dfrac{\omega e}{p^{2}a^{3}k},\\ \\ -2\mathrm{C}\mathrm{E}_{0}(i\lambda a)-2\mathrm{DS}_{2}\left(\lambda_{1}a\right)+\dfrac{i\mathrm{B}}{pa^{3}}=0. \end{array}\right\} $$}}

Since, on the assumption discussed above, the electrification on the surface of the moving sphere is equivalent to a tangential current-sheet whose intensity is $$\omega\epsilon^{ipt}\sin\theta\dfrac{e}{4\pi a^{2}}$$, we have as another surface-condition that the difference between the