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

CHAPTER VIII.

Simple Cases of Electrification.

Art.

Page

124. Two parallel planes

150

125. Two concentric spherical surfaces

152

126. Two coaxal cylindric surfaces

154

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

155

CHAPTER IX.

Spherical Harmonics.

128. Singular points at which the potential becomes infinite

157

129. Singular points of different orders defined by their axes

158

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

160

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

162

132. The zonal, tesseral, and sectorial types

163

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

165

134. Application to the theory of electrified spherical surfaces

166

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

167

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

169

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

169

138. On conjugate harmonics

170

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 $Y_{j}$

171

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

172

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

173