Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/211

integral of $VY_{i}dS$, extended over every element $dS_{1}$ of the surface of a sphere of radius $a$, is given by the equation

$\iint VY_{i}dS={\frac {4\pi }{|{\underline {i}}}}{\frac {a^{i+2}}{2i+1}}{\frac {d^{i}V}{dh_{1}\dots dh_{1}}};$

(52)

where the differentiations of $V$ are taken with respect to the axes of the harmonic $Y_{i}$, and the value of the differential coefficient is that at the centre of the sphere.

136.] Let us now suppose that $V$ is a solid harmonic of positive degree $j$ of the form j

$V={\frac {r^{j}}{a^{j}}}Y_{j}.$

(53)

At the spherical surface, $r=a$, the value of $V$ is the surface harmonic $Y_{j}$, and equation (52) becomes

$\iint Y_{i}Y_{j}dS={\frac {4\pi }{|{\underline {i}}}}{\frac {a^{i-j+2}}{2i+1}}{\frac {d^{i}\left(r^{j}Y_{j}\right)}{dh_{1}\dots dh_{i}}},$

(54)

where the value of the differential coefficient is that at the centre of the sphere.

When $i$ is numerically different from $j$, the surface-integral of the product $Y_{i}Y_{j}$ vanishes. For, when $i$ is less than $j$, the result of the differentiation in the second member of (54) is a homogeneous function of x, y, and z, of degree $j-i$, the value of which at the centre of the sphere is zero. If $i$ is equal to $j$ the result is a constant, the value of which will be determined in the next article. If the differentiation is carried further, the result is zero. Hence the surface-integral vanishes when $i$ is greater than $j$.

137.] The most important case is that in which the harmonic $r^{j}Y_{j}$ is differentiated with respect to $i$ new axes in succession, the numerical value of $j$ being the same as that of $i$, but the directions of the axes being in general different. The final result in this case is a constant quantity, each term being the product of $i$ cosines of angles between the different axes taken in pairs. The general form of such a product may be written symbolically

$\mu _{ii}^{s}\mu _{jj}^{s}\mu _{ij}^{i-2s},$

which indicates that there are $s$ cosines of angles between pairs of axes of the first system and $s$ between axes of the second system, the remaining $i-2s$ cosines being between axes one of which belongs to the first and the other to the second system.

In each product the suffix of every one of the $2i$ axes occurs once, and once only.