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

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

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

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

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.