Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/261

${\begin{array}{l}{\frac {dq_{aa}}{dc}}=-{\frac {2a^{2}bc}{\left(c^{2}-b^{2}\right)^{2}}}-{\frac {2a^{3}b^{2}c\left(2c^{2}-2b^{2}-a^{2}\right)}{\left(c^{2}-b^{2}+ac\right)^{2}\left(c^{2}-b^{2}-ac\right)^{2}}}-etc.,\\\\{\frac {dq_{ab}}{dc}}={\frac {ab}{c^{2}}}+{\frac {a^{2}b^{2}\left(3c^{2}-a^{2}-b^{2}\right)}{c^{2}\left(c^{2}-a^{2}-b^{2}\right)}}+{\frac {a^{3}b^{3}\left\{\left(5c^{2}-a^{2}-b^{2}\right)\left(c^{2}-a^{2}-b^{2}\right)-a^{2}b^{2}\right\}}{c^{2}\left(c^{2}-a^{2}-b^{2}+ab\right)^{2}\left(c^{2}-a^{2}-b^{2}-ab\right)^{2}}}-etc.,\\\\{\frac {dq_{bb}}{dc}}=-{\frac {2ab^{2}c}{\left(c^{2}-b^{2}\right)^{2}}}-{\frac {2a^{2}b^{3}c\left(2c^{2}-2a^{2}-b^{2}\right)}{\left(c^{2}-a^{2}+bc\right)^{2}\left(c^{2}-a^{2}-bc\right)^{2}}}-etc.\end{array}}$

Distribution of Electricity on Two Spheres in Contact.

175.] If we suppose the two spheres at potential unity and not influenced by any other point, then, if we invert the system with respect to the point of contact, we shall have two parallel planes, distant ${\tfrac {1}{2a}}$ and ${\tfrac {1}{2b}}$ from the point of inversion, and electrified by the action of a unit of electricity at that point.

There will be a series of positive images, each equal to unity, at distances $s\left({\tfrac {1}{a}}+{\tfrac {1}{b}}\right)$ from the origin, where $s$ may have any integer value from $-\infty$ to $+\infty$.

There will also be a series of negative images each equal to -1, the distances of which from the origin, reckoned in the direction of $a$, are ${\tfrac {1}{a}}+s\left({\tfrac {1}{a}}+{\tfrac {1}{b}}\right)$.

When this system is inverted back again into the form of the two spheres in contact, we have a corresponding series of negative images, the distances of which from the point of contact are of the form ${\tfrac {1}{s\left({\frac {1}{a}}+{\frac {1}{b}}\right)}}$, where $s$ is positive for the sphere $A$ and negative for the sphere $B$. The charge of each image, when the potential of the spheres is unity, is numerically equal to its distance from the point of contact, and is always negative.

There will also be a series of positive images whose distances from the point of contact measured in the direction of the centre of $a$, are of the form ${\tfrac {1}{{\frac {1}{a}}+s\left({\frac {1}{a}}+{\frac {1}{b}}\right)}}$.

When $s$ is zero, or a positive integer, the image is in the sphere $A$.

When $s$ is a negative integer the image is in the sphere $B$.