Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/214



where $$Y_{i(j)}$$ denotes the value of $$Y_i$$ in equation (27) at the common pole of all the axes of $$Q_j$$.

140.] This result is a very important one in the theory of spherical harmonics, as it leads to the determination of the form of a series of spherical harmonics, which expresses a function having any arbitrarily assigned value at each point of a spherical surface.

For let $$F$$ be the value of the function at any given point of the sphere, say at the centre of gravity of the element of surface $$dS$$, and let $$Q_i$$ be the zonal harmonic of degree $$i$$ whose pole is the point $$P$$ on the sphere, then the surface-integral

extended over the spherical surface will be a spherical harmonic of degree $$i$$, because it is the sum of a number of zonal harmonics whose poles are the various elements $$dS$$, each being multiplied by $$FdS$$. Hence, if we make

we may expand F in the form

or

This is the celebrated formula of Laplace for the expansion in a series of spherical harmonics of any quantity distributed over the surface of a sphere. In making use of it we are supposed to take a certain point $$P$$ on the sphere as the pole of the zonal harmonic $$Q_i$$, and to find the surface-integral

over the whole surface of the sphere. The result of this operation when multiplied by $$2i+1$$ gives the value of $$A_{i}Y_{i}$$ at the point $$P$$, and by making $$P$$ travel over the surface of the sphere the value of $$A_{i}Y_{i}$$ at any other point may be found.