Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/26

Chapter VIII.

Simple Cases of Electrification.

Art.

124. Two parallel planes

125. Two concentric spherical surfaces

126. Two coaxal cylindric surfaces

127. Longitudinal force on a cylinder, the ends of which are surrounded by cylinders at different potentials

Chapter IX.

Spherical Harmonics.

128. Singular points at which the potential becomes infinite

129. Singular points of different orders defined by their axes

130. Expression for the potential due to a singular point referred to its axes

131. This expression is perfectly definite and represents the most general type of the harmonic of $i$ degrees

132. The zonal, tesseral, and sectorial types

133. Solid harmonics of positive degree. Their relation to those of negative degree

134. Application to the theory of electrified spherical surfaces

135. The external action of an electrified spherical surface compared with that of an imaginary singular point at its centre

136. Proof that if $Y_{i}$ and $Y_{j}$ are two surface harmonics of different degrees, the surface-integral $\iint Y_{i}Y_{j}dS=0$, the integration being extended over the spherical surface

137. Value of $\iint Y_{i}Y_{j}dS$ where $Y_{i}$ and $Y_{j}$ are surface harmonics of the same degree but of different types

138. On conjugate harmonics

139. If $Y_{j}$ is the zonal harmonic and $Y_{i}$ any other type of the same degree

$\iint Y_{i}Y_{j}dS={\frac {4\pi a^{2}}{2i+1}}Y_{i(j)}$

where $Y_{i(j)}$ is the value of $Y_{i}$ at the pole of $Y_{j}$

140. Development of a function in terms of spherical surface harmonics

141. Surface-integral of the square of a symmetrical harmonic