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A Treatise on Electricity and Magnetism, Vol. I (1873) - Page 27 - Source Library

A Treatise on Electricity and Magnetism, Vol. I

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English

CHAPTER VIII.

SIMPLE CASES OF ELECTRIFICATION.

Article		Page
124.	Two parallel planes .. .. .. .. .. .. .. ..	150
125.	Two concentric spherical surfaces .. .. .. .. .. ..	152
126.	Two coaxial original: "coaxal" cylindrical surfaces .. .. .. .. .. ..	154
127.	Longitudinal force on a cylinder, the ends of which are surrounded by cylinders at different potentials .. .. .. ..	155

CHAPTER IX.

SPHERICAL HARMONICS.

Article		Page
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 These are specific mathematical patterns used to describe distributions on the surface of a sphere. .. .. .. ..	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 center .. ..	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 $\iint Y_i Y_j dS = \frac{4 π a^2}{2i + 1} Y_{i(j)}$ 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

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