Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/260

The repulsion between the spheres is therefore, by Arts. 92, 93,

${\begin{array}{ll}F=&{\frac {1}{2}}\left\{V_{a}^{2}{\frac {dq_{aa}}{dc}}+2V_{a}V_{b}{\frac {dq_{ab}}{dc}}+V_{b}^{2}{\frac {dq_{bb}}{dc}}\right\},\\\\=-&{\frac {1}{2}}\left\{E_{a}^{2}{\frac {dp_{aa}}{dc}}+2E_{a}E_{b}{\frac {dp_{ab}}{dc}}+E_{b}^{2}{\frac {dp_{bb}}{dc}}\right\},\end{array}}$

where $c$ is the distance between the centres of the spheres.

Of these two expressions for the repulsion, the first, which expresses it in terms of the potentials of the spheres and the variations of the coefficients of capacity and induction, is the most convenient for calculation.

We have therefore to differentiate the $q$'s with respect to $c$. These quantities are expressed as functions of $k,\alpha ,\beta$, and $\varpi$, and must be differentiated on the supposition that $a$ and $b$ are constant. From the equations

$k=a\sin ha=b\sin h\beta =c{\frac {\sin ha\sin h\beta }{\sin h\varpi }},$

we find

${\begin{array}{l}{\frac {d\alpha }{dc}}={\frac {\sin ha\cos h\beta }{k\sin h\varpi }},\\\\{\frac {d\beta }{dc}}={\frac {\sin ha\sin h\beta }{k\sin h\varpi }},\\\\{\frac {d\varpi }{dc}}={\frac {1}{k}},\\\\{\frac {dk}{dc}}={\frac {\cos ha\cos h\beta }{\sin h\varpi }};\end{array}}$

whence we find

${\begin{array}{l}{\frac {dq_{aa}}{dc}}={\frac {\cos ha\cos h\beta }{\sin h\varpi }}{\frac {q_{aa}}{k}}-\sum _{s=0}^{s=\infty }{\frac {(sc-a\cos h\beta )\cos h(s\varpi -\alpha )}{c(\sin h(s\varpi -\alpha ))^{2}}},\\\\{\frac {dq_{ab}}{dc}}={\frac {\cos ha\cos h\beta }{\sin h\varpi }}{\frac {q_{ab}}{k}}+\sum _{s=1}^{s=\infty }{\frac {s\cos hs\varpi }{(\sin hs\varpi )^{2}}},\\\\{\frac {dq_{bb}}{dc}}={\frac {\cos ha\cos h\beta }{\sin h\varpi }}{\frac {q_{bb}}{k}}-\sum _{s=0}^{s=\infty }{\frac {(sc+b\cos h\alpha )\cos h(\beta +s\varpi )}{c(\sin h(\beta +s\varpi ))^{2}}}.\end{array}}$

Sir William Thomson has calculated the force between two spheres of equal radius separated by any distance less than the diameter of one of them. For greater distances it is not necessary to use more than two or three of the successive images.

The series for the differential coefficients of the $q$'s with respect to $c$ are easily obtained by direct differentiation