1911 Encyclopædia Britannica/Spherical Harmonics

by Ernest William Hobson

SPHERICAL HARMONICS, in mathematics, certain functions of fundamental importance in the mathematical theories of gravitation, electricity, hydrodynamics, and in other branches of physics. The term “spherical harmonic” is due to Lord Kelvin, and is primarily employed to denote either a rational integral homogeneous function of three variables x, y, z, which satisfies the differential equation

$$\nabla ^{2}V={\frac {\partial ^{2}V}{\partial x^{2}}}+{\frac {\partial ^{2}V}{\partial y^{2}}}+{\frac {\partial ^{2}V}{\partial z^{2}}}=0,$$

known as Laplace's equation, or a function which satisfies the differential equation, and becomes a rational integral homogeneous function when multiplied by a power of $(x^{2}+y^{2}+z^{2})^{\frac {1}{2}}$. Of all particular integrals of Laplace's equation, these are of the greatest importance in respect of their applications, and were the only ones considered by the earlier investigators; the solutions of potential problems in which the bounding surfaces are exactly or approximately spherical are usually expressed as series in which the terms are these spherical harmonics. In the wider sense of the term, a spherical harmonic is any homogeneous function of the variables which satisfies Laplace's equation, the degree of the function being not necessarily integral or real, and the functions are not necessarily rational in x, y, z, or single-valued; the functions may, when necessary, be termed ordinary spherical harmonics. For the treatment of potential problems which relate to spaces bounded by special kinds of surfaces, solutions of Laplace's equation are required which are adapted to the particular boundaries, and various classes of such solutions have thus been introduced into analysis. Such functions are usually of a more complicated structure than ordinary spherical harmonics, although they possess analogous properties. As examples we may cite Bessel's functions in connexion with circular cylinders, Lamé's functions in connexion with ellipsoids, and toroidal functions for anchor rings. The theory of such functions may be regarded as embraced under the general term harmonic analysis. The present article contains an account of the principal properties of ordinary spherical harmonics, and some indications of the nature and properties of the more important of the other classes of functions which occur in harmonic analysis. Spherical and other harmonic functions are of additional importance in view of the fact that they are largely employed in the treatment of the partial differential equations of physics, other than Laplace's equation; as examples of this, we may refer to the equation ${\frac {\partial u}{\partial t}}=k\nabla ^{2}u$, which is fundamental in the theory of conduction of heat and electricity, also to the equation ${\frac {\partial ^{2}u}{\partial t^{2}}}=k\nabla ^{2}u$, which occurs in the theory of the propagation of aerial and electro-magnetic waves. The integration under given conditions of more complicated equations which occur in the theories of hydro-dynamics and elasticity, can in certain cases be effected by the use of the functions employed in harmonic analysis.

1. Relation between Spherical Harmonics of Positive and Negative Degrees.—A function which is homogeneous in x, y, z, of degree n in those variables, and which satisfies Laplace's equation

${\frac {\partial ^{2}V}{\partial x^{2}}}+{\frac {\partial ^{2}V}{\partial y^{2}}}+{\frac {\partial ^{2}V}{\partial z^{2}}}=0$, or $\nabla ^{2}V=0$,

is termed a solid spherical harmonic, or simply a spherical harmonic of degree n. The degree n may be fractional or imaginary, but we are at present mainly concerned with the case in which n is a positive or negative integer. If x, y, z be replaced by their values $r\sin \theta \cos \phi$, $r\sin \theta \sin \phi$, $r\cos \theta$ in polar co-ordinates, a solid spherical harmonic takes the form $r^{n}f_{n}(\theta ,\phi )$; the factor $f_{n}(\theta ,\phi )$ is called a surface harmonic of degree n. If V_n denote a spherical harmonic of degree n, it may be shown by differentiation that $\nabla ^{2}(r^{m}V_{n})=m(2n+m+1)r^{m-2}V_{n}$, and thus as a particular case that $\nabla ^{2}(r^{-2n-1}V_{n})=0$; we have thus the fundamental theorem that from any spherical harmonic V_n of degree n, another of degree $-n-1$ may be derived by dividing V_n by $r^{2n+1}$. All spherical harmonics of negative integral degree are obtainable in this way from those of positive integral degree. This theorem is a particular case of the more general inversion theorem that if $F(x,y,z)$ is any function which satisfies the equation (1), the function

$${\frac {1}{r}}F\left({\frac {x}{r^{2}}},{\frac {y}{r^{2}}},{\frac {z}{r^{2}}}\right)$$

also satisfies the equation.

The ordinary spherical harmonics of positive integral degree n are those which are rational integral functions of x, y, z. The most general rational integral function of degree n in three letters contains ${\frac {1}{2}}(n+1)(n+2)$ coefficients; if the expression be substituted in (1), we have on equating the coefficients separately to zero ${\frac {1}{2}}n(n-1)$ relations to be satisfied; the most general spherical harmonic of the prescribed type therefore contains ${\frac {1}{2}}(n+1)(n+2)-{\frac {1}{2}}n(n-1)$, or $2n+1$ independent constants. There exist, therefore, $2n+1$ independent ordinary harmonics of degree n; and corresponding to each of these there is a negative harmonic of degree $-n-1$ obtained by dividing by $r^{2n+1}$. The three independent harmonics of degree 1 are x, y, z; the five of degree 2 are $y^{2}-z^{2}$, $z^{2}-x^{2}$, $yz$, $zx$, $xy$. Every harmonic of degree n is a linear function of $2n+1$ independent harmonics of the degree; we proceed, therefore, to find the latter.

2. Determination of Harmonics of given Degree.—It is clear that a function $f(ax+by+cz)$ satisfies the equation (1), if a, b, c are constants which satisfy the condition $a^{2}+b^{2}+c^{2}=0$; in particular the equation is satisfied by $(z+\iota x\cos \alpha +\iota y\sin \alpha )^{n}$. Taking n to be a positive integer, we proceed to expand this expression in a series of cosines and sines of multiples of $\alpha$; each term will then satisfy (1) separately. Denoting $e^{\iota \alpha }$ by k, and $y+\iota x$ by t, we have

$$(z+\iota x\cos \alpha +\iota y\sin \alpha )^{n}=\left(z+{\frac {1}{2}}kt+{\frac {z^{2}}{2kt}}-{\frac {r^{2}}{2kt}}\right)^{n}$$

which may be written as $(2kt)^{-n}\{(z+kt)^{2}-r^{2}\}^{n}$. On expansion by Taylor's theorem this becomes

$$(2kt)^{-n}\sum _{0}^{2n}{\frac {k^{3}t^{3}}{s!}}{\frac {\partial ^{3}}{\partial z^{3}}}(z^{2}-r^{2})^{n},$$

the differentiation applying to z only as it occurs explicitly; the terms involving $\cos m\alpha$, $\sin m\alpha$ in this expansion are

$${\frac {1}{2^{n}}}\cos m\alpha \left\{{\frac {(y+ix)^{m}}{(n+m)!}}{\frac {\partial ^{n+m}}{\partial z^{n+m}}}(z^{2}-r^{2})^{n}+{\frac {(y+ix)^{-m}}{(n+m)!}}{\frac {\partial ^{n-m}}{\partial z^{n-m}}}(z^{2}-r^{2})^{m}\right\}$$

$$i{\frac {1}{2^{n}}}\sin m\alpha \left\{{\frac {(y+ix)^{m}}{(n+m)!}}{\frac {\partial ^{n+m}}{\partial z^{n+m}}}(z^{2}-r^{2})^{n}+{\frac {(y+ix)^{-m}}{(n-m)!}}{\frac {\partial ^{n-m}}{\partial z^{n-m}}}(z^{2}-r^{2})^{m}\right\}$$

where $m=1,2,\dots ,n$; and the term independent of $\alpha$ is

$${\frac {1}{2^{n}n!}}{\frac {\partial ^{n}}{\partial z^{n}}}(z^{2}-r^{2})^{n}.$$

On writing $$(y+ix)^{m}=i^{m}r^{m}(\cos m\phi -i\sin m\phi )2\sin ^{m}\theta ,(y+ix)^{-m}=i^{-m}r^{-m}(\cos m\phi +i\sin m\phi )2\sin ^{-m}\theta$$

and observing that in the expansion of $(z+\iota x\cos \alpha +\iota y\sin \alpha )^{m}$ the expressions $\cos m\alpha ,\sin m\alpha$ can only occur in the combination $\cos m(\phi -\alpha )$, we see that the relation

$$i^{m}r^{m}{\frac {\sin ^{m}\theta }{(n+m)!}}{\frac {\partial ^{n+m}}{\partial z^{n+m}}}(z^{2}-r^{2})^{n}=i^{-m}r^{-m}{\frac {\sin ^{-m}\theta }{(n-m)!}}{\frac {\partial ^{n-m}}{\partial z^{n-m}}}(z^{2}-r^{2})^{n}$$

must hold identically, and thus that the terms in the expansion reduce to

$${\frac {1}{(n+m)!}}{\frac {i^{m}}{2^{n-1}}}r^{m}\cos m\alpha \cos m\phi \sin ^{m}\theta {\frac {\partial ^{n+m}}{\partial z^{n+m}}}(z^{2}-r^{2})^{n}$$

$${\frac {1}{(n+m)!}}{\frac {i^{m}}{2^{n-1}}}r^{m}\sin m\alpha \sin m\phi \sin ^{m}\theta {\frac {\partial ^{n+m}}{\partial z^{n+m}}}(z^{2}-r^{2})^{n}$$

We thus see that the spherical harmonics of degree n are of the form

$$r^{n}~{\stackrel {\cos }{\sin }}~m\phi \sin ^{m}\theta {\frac {d^{n+m}}{d\mu ^{n+m}}}(\mu ^{2}-1)^{n}$$

where $\mu$ denotes $\cos \theta$; by giving m the values $0,1,\dots ,n$ we thus have the $2n+1$ functions required. On carrying out the differentiations we see that the required functions are of the form

$${\begin{aligned}&A[(x+iy)^{m}=(x-iy)^{m}]{\Bigl \{}z^{n-m}-{\frac {(n-m)(n-m-1)}{2.2n-1}}z^{n-m-2}(x^{2}+y^{2}+z^{2})\\&+{\frac {(n-m)(n-m-1)(n-m-2)(n-m-3)}{2.4.2n-1.2n-3}}z^{n-m-4}(x^{2}+y^{2}+z^{2})^{2}+\cdots {\Bigr \}}\end{aligned}}$$

where $m=0,1,2,3,\dots ,n$.

3. Zonal, Tesseral and Sectorial Harmonics.–Of the system of $2n+1$ harmonics of degree n, only one is symmetrical about the z axis; this is

$$r^{n}{\frac {1}{2^{n}n!}}{\frac {d^{n}}{d\mu ^{n}}}(\mu ^{2}-1)^{n};$$

writing

$$P_{n}(\mu )={\frac {1}{2^{n}n!}}{\frac {d^{n}}{d\mu ^{n}}}(\mu ^{2}-1)^{n},$$

we observe that $P_{n}(\mu )$ has n zeros all lying between =1, consequently the locus of points on a sphere $r=a$, for which $P_{n}(\mu )$ vanishes is n circles all parallel to the meridian plane: these circles divide the sphere into zones, thus $P_{n}(\mu )$ is called the zonal surface harmonic of degree n, and $r^{n}P_{n}(\mu ),r^{-n-1}P_{n}(\mu )$ are the solid zonal harmonics of degrees n and $-n-1$. The locus of points on a sphere for which ${\stackrel {\cos }{\sin }}~m\phi .\sin ^{m}\theta {\frac {d^{n+m}}{d\mu ^{n+m}}}(\mu ^{2}-1)^{n}$ vanishes consists of $n-m$ circles parallel to the meridian plane, and m great circles through the poles; these circles divide the spherical surface into quadrilaterals or τέσσερα, except when $n=m$, in which case the surface is divided into sectors, and the harmonics are therefore called tesseral, except those for which $m=n$, which are called sectorial. Denoting $(1-\mu ^{2}){\frac {{\frac {1}{2}}^{m}d^{m}P_{n}(\mu )}{d\mu ^{m}}}$ by $P_{n}^{m}(\mu )$, the tesseral surface harmonics are ${\stackrel {\cos }{\sin }}~m\phi .P_{n}^{m}(\cos \theta )$, where $m=1,2,\dots ,n-1$, and the sectorial harmonics are ${\stackrel {\cos }{\sin }}~n\phi .P_{n}^{n}(\cos \theta )$. The functions $P_{n}(\mu ),P_{n}^{m}(\mu )$ denote the expressions $$P_{n}(\mu )={\frac {(2n)!}{2^{n}n!n!}}\left\{\mu ^{n}-{\frac {n(n-1)}{2.2n-1}}\mu ^{n-2}+{\frac {n(n-1)(n-2)(n-3)}{2.4.2n-1.2n-3}}\mu ^{n-4}\dots \right\}$$

$${\begin{aligned}P_{n}^{m}(\mu )={\frac {(2n)!}{2^{n}n!(n-m)!}}(1-\mu ^{2})^{{\frac {1}{2}}m}&{\Bigl \{}\mu ^{n-m}\\&-{\frac {(n-m)(n-m-1)}{2.2n-1}}\mu ^{n-m-2}+\dots {\Bigr \}}\end{aligned}}$$

$$P_{n}^{n}(\mu )={\frac {(2n)!}{2^{n}n!}}(1-\mu ^{2})^{{\frac {1}{2}}n}.$$

Every ordinary harmonic of degree n is expressible as a linear function of the system of $2n+1$ zonal, tesseral and sectorial harmonics of degree n; thus the general form of the surface harmonic is $$a_{0}P_{n}(\mu )+\sum _{m=1}^{n}(a_{m}\cos m\phi +b_{m}\sin m\phi )P_{n}^{m}(\mu )$$

In the present notation we have

$$(z+\iota x\cos \alpha +\iota y\sin \alpha )^{n}=r^{n}\left\{P_{n}(\mu )+2\sum _{1}^{n}\iota ^{m}{\frac {n!}{(n+m)!}}P_{n}^{m}(\mu )\cos m(\phi -\alpha )\right\}$$

if we put $\alpha =0$, we thus have

$$(\cos \theta +\iota \sin \theta \cos \phi )^{n}=P_{n}(\cos \theta )+2\sum _{1}^{n}i^{m}{\frac {n!}{(n+m!)}}P_{n}^{m}(\cos \theta )\cos m\phi ,$$

from this we obtain expressions for $P_{n}(\cos \theta ),P_{n}^{m}(\cos \theta )$ as definite integrals

$$P_{n}(\cos \theta )={\frac {1}{\pi }}\int _{0}^{\pi }(\cos \theta +\iota \sin \theta \cos \phi )^{n}d\phi$$

$$\iota ^{m}{\frac {n!}{(n+m)!}}P_{n}^{m}(\cos \theta )={\frac {1}{\pi }}\int _{0}^{\pi }(\cos \theta +\iota \sin \theta \cos \phi )^{n}\cos m\phi d\phi .$$

4. Derivation of Spherical Harmonics by Differentiation.–The linear character of Laplace's equation shows that, from any solution, others may be derived by differentiation with respect to the variables x, y, z; or, more generally, if

$$f\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)$$

denote any rational integral operator,

$$f\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)V$$

is a solution to the equation, if V satisfies it. This principle has been applied by Thomson and Tait to the derivation of the system of any integral degree, by operating upon $1/r$, which satisfies Laplace's equation. The operations may be conveniently carried out by means of the following differentiation theorem. (See papers by Hobson, in the Messenger of Mathematics, xxiii. 115, and Proc. Lond. Math. Soc., vol. xxiv.)

$${\begin{aligned}f_{n}\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right){\frac {1}{r}}&=(-1)^{n}{\frac {(2n)!}{2^{n}n!}}{\frac {1}{r^{2n+1}}}{\Bigl \{}1-{\frac {r^{2}\nabla ^{2}}{2.2n-1}}\\&+{\frac {r^{4}\nabla ^{4}}{2.4.2n-1.2n-3}}-\dots {\Bigr \}}f_{n}(x,y,z)\end{aligned}}$$

which is a particular case of the more general theorem

$${\begin{aligned}f_{n}\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)F(r)&={\Bigl \{}2^{n}{\frac {d^{n}F}{d(r^{2})^{n}}}+{\frac {2^{n-2}}{1!}}{\frac {d^{n-1}F}{d(r^{2})^{n-1}}}\nabla ^{2}+\dots \\&+{\frac {2^{n-2s}}{s!}}{\frac {d^{n-s}F}{d(r^{2})^{n-s}}}\nabla ^{2s}+\dots {\Bigr \}}f_{n}(x,y,z)\end{aligned}}$$

where $f_{n}(x,y,z)$ is a rational integral homogeneous function of degree n. The harmonic of positive degree n corresponding to that of degree $-n-1$ in the expression (7) is

$$\left\{1-{\frac {r^{2}\nabla ^{2}}{2.2n-1}}+{\frac {r^{4}\nabla ^{4}}{2.4.2n-1.2n-3}}-\dots \right\}f_{n}(x,y,z).$$

It can be verified that even when n is unrestricted, this expression satisfies Laplace's equation, the sole restriction being that of the convergence of the series.

5. Maxwell's Theory of Poles.—Before proceeding to obtain by means of (7), the expressions for the zonal, tesseral, and sectorial harmonics, it is convenient to introduce the conception, due to Maxwell (see Electricity and Magnetism, vol. i. ch. ix.), of the poles of a spherical harmonic. Suppose a sphere of any radius drawn with its centre at the origin; any line whose direction-cosines are l, m, n drawn from theo rigin, is called an axis, and the point where this axis cuts the sphere is called the pole of the axis. Different axes will be denoted by suffixes attached to the direction-cosines: the cosine $(l_{\iota }x+m_{\iota }y+n_{\iota }z)/r$ of the angle between the radius vector r to a point $(x,y,z)$ and the axis $(l_{\iota },m_{\iota },n_{\iota })$ will be denoted by $\lambda _{\iota }$; the cosine of the angle between two axes is $l_{\iota }l_{\gamma }+m_{\iota }m_{\gamma }+n_{\iota }n_{\gamma }$, which will be noted by $\mu _{\iota \gamma }$. The operation

$$l_{\iota }{\frac {\partial }{\partial x}}+m_{\iota }{\frac {\partial }{\partial y}}+n_{\iota }{\frac {\partial }{\partial z}}$$

performed upon any function of x, y, z, is spoken of as differentiation with respect to the axis $(l_{\iota },m_{\iota },n_{\iota })$, and is denoted by $\partial /\partial h_{\iota }$. The potential function $V_{0}=e_{0}/r$ is defined to be the potential due to a singular point of degree zero at the origin; $e_{0}$ is called the strength of the singular point. Let a singular point of degree zero, and strength $e_{0}$, be on an axis $h_{1}$, at a distance $a_{0}$from the origin, and also suppose that the origin is a singular point of strength $-e_{0}$; let $e_{0}$ be indefinitely increased, and $a_{0}$ indefinitely diminished, but so that the product $e_{0}a_{0}$ is finite and equal to $e_{0}$; the origin is then said to be a singular point of the first degree, of strength $e_{1}$, the axis being $h_{1}$. Such a singular point is frequently called a doublet. In a similar manner, by placing two singular points of degree, unity and strength, $e_{1}$, $-e_{1}$, at a distance $a_{1}$ along an axis $h_{2}$, and at the origin respectively, when $e_{1}$ is indefinitely increased, and $a_{1}$ diminished so that $e_{1}a_{1}$ is finite and $=e_{2}$, we obtain a singular point of degree 2, strength $e_{2}$ at the origin, the axes being $h_{1},h_{2}$. Proceeding in this manner we arrive at the conception of a singular point of any degree n, of strength $e_{n}$ at the origin, the singular point having any n given axes $h_{1},h_{2},\dots h_{n}$. If $e_{n-1}\phi _{n-1}(x,y,z)$ is the potential due to a singular point at the origin, of degree $n-1$, and strength $e_{n-1}$, with axes $h_{1},h_{2},\dots h_{n-1}$, the potential of a singular point of degree n, the new axis of which is $h_{n}$ is the limit of

$$e_{n-1}\phi _{n-1}(x-l_{n}a,y-m_{n}a,z-n_{n}a)-e_{n-1}\phi _{n-1}(x,y,z)$$

when

$$La=0,Le_{n-1}=\infty ,Lae_{n-1}=e_{n};$$

this limit is

$-e_{n}(l_{n}{\frac {\partial \phi _{n-1}}{\partial x}}+m_{n}{\frac {\partial \phi _{n-1}}{\partial y}}+n_{n}{\frac {\partial \phi _{n-1}}{\partial z}})$, or $-e_{n}{\frac {\partial }{\partial h_{n}}}\phi _{n-1}$.

Since $V_{0}=1/r$, we see that the potential V, due to a singular point at the origin of strength $e_{n}$ and axes $h_{1},h_{2},\dots h_{n}$ is given by

$$V_{n}=(-1)^{n}e_{n}{\frac {\partial ^{n}}{\partial h_{1}\partial h_{2}\dots \partial h_{n}}}{\frac {1}{r}}.$$

6. Expression for a Harmonic with given Poles.—The result of performing the operations in (8) is that $V_{n}$ is of the form

$$n!e_{n}{\frac {Y_{n}}{r^{n+1}}},$$

where $Y_{n}$ is a surface harmonic of degree n, and will appear as a function of the angles which r makes with the n axes, and of the angles these axes make with one another. The poles of the n axes are defined to be the poles of the surface harmonics, and are also frequently spoken of as the poles of the solid harmonics $Y_{n}r^{n},Y_{n}r^{-n-1}$. Any spherical harmonic is completely specified by means of its poles.

In order to express $Y_{n}$ in terms of the positions of its poles, we apply the theorem (7) to the evaluation of $Y_{n}$ in (8). On putting

$f_{n}(x,y,z)=\prod _{r=1}^{r=n}(l_{r}x+m_{r}y+n_{r}z)$, we have

$$Y_{n}={\frac {(2n!)}{2^{n}n!n!}}\cdot {\frac {1}{r^{n}}}\left(1-{\frac {r^{2}\nabla ^{2}}{2.2n-1}}+{\frac {r^{4}\nabla ^{4}}{2.4.2n-1.2n-3}}\dots \right)\times \prod _{r}^{n}(l_{r}x+m_{r}y+n_{r}z).$$

By $\sum (\mu ^{s}\lambda ^{n-2s})$ we shall denote the sum of the products of $s$ of the quantities $\mu$, and $n-2s$ of the quantities $\lambda$; in any term each suffix is to occur once, and once only, every possible order being taken. We find

$$\prod (lx+my+nz)=\sum (\lambda ^{n})r^{n}\Delta ^{2}\prod (lx+my+nz)=2\sum (\mu ^{1}\lambda ^{n-2})r^{n-2},$$

and generally

$$\Delta ^{2m}\prod (lx+my+nz)=2^{m}m!\sum (\mu ^{m}\lambda ^{n-2m})r^{n-2m};$$

thus we obtain the following expression for $Y_{n}$, the surface harmonic which has given poles $h_{1},h_{2},\dots h_{n}$;

$${\begin{aligned}Y_{n}&=r^{n+1}{\frac {(-1)^{n}}{n!}}{\frac {\partial ^{n}}{\partial h_{1}\partial h_{2}\dots \partial h_{n}}}\cdot {\frac {1}{r}}\\&={\underset {m=0}{S}}\left\{(-1)^{m}{\frac {(2n-2m)!}{2^{n-m}n!(n-m)!}}\sum (\lambda ^{n-2m}\mu ^{m})\right\}\end{aligned}}$$

where S denotes a summation with respect to m from $m=0$ to $m={\frac {1}{2}}n$, or ${\frac {1}{2}}(n-1)$, according as n is even or odd. This is Maxwell's general expression (loc. cit.) for a surface harmonic with given poles.

If the poles on a sphere of radius r are denoted by A, B, C. . ., we obtain from (9) the following expressions for the harmonics of the first four degrees:—

$$Y_{1}=\cos PA,Y_{2}={\frac {1}{2}}(3\cos PA\cos PB-\cos AB),$$

$${\begin{aligned}Y_{3}={\frac {1}{2}}(15\cos PA\cos PB\cos PC&-\cos PA\cos BC-\cos PB\cos CA\\&-\cos PC\cos AB),\end{aligned}}$$

$${\begin{aligned}Y_{4}={\frac {1}{8}}(35\cos PA\cos PB\cos PC\cos PD&-5\sum \cos PA\cos PB\cos CD\\&+\sum \cos AB\cos CD).\end{aligned}}$$

7. Poles of Zonal, Tesseral and Sectorial Harmonics.—Let the n axes of the harmonic coincide with the axis of z, we have then by (8) the harmonic

$${\frac {(-1)^{n}r^{n+1}}{n!}}{\frac {\partial ^{n}}{\partial z^{n}}}{\frac {1}{r}}$$

applying the theorem (7) to evaluate this expression, we have

$${\begin{aligned}{\frac {(-1)^{n}r^{n+1}}{n!}}{\frac {\partial ^{n}}{\partial z^{n}}}{\frac {1}{r}}&={\frac {(2n)!}{2^{n}n!n!}}{\frac {1}{r^{n}}}\left\{1-{\frac {r^{2}\nabla ^{2}}{2.2n-1}}+{\frac {r^{4}\nabla ^{4}}{2.4.2n-1.2n-3}}-\right\}z^{n}\\&={\frac {(2n)!}{2^{n}n!n!}}\left\{\mu ^{n}-{\frac {n(n-1)}{2.2n-1}}\mu ^{n-2}+\dots \right\},\end{aligned}}$$

the expression on the right side is $P^{n}(\mu )$, the zonal surface harmonic; we have therefore

$$P_{n}(\mu )={\frac {(-1)^{n}r^{n+1}}{n!}}{\frac {\partial ^{n}}{\partial z^{n}}}{\frac {1}{r}}.$$

The zonal harmonic has therefore all its poles coincident with the z axis. Next, suppose n - m axes coincide with the z axis, and that the remaining m axes are distributed symmetrically in the plane of x, y at intervals $\pi /m$, the direction cosines of one of them being $\cos \alpha ,\sin \alpha ,0$. We have

$${\begin{aligned}\prod _{0}^{m-1}\left\{\cos \left(\alpha +{\frac {r\pi }{m}}\right){\frac {\partial }{\partial x}}+\sin \left(\alpha +{\frac {r\pi }{m}}\right){\frac {\partial }{\partial y}}\right\}&={\frac {1}{2^{m}}}\prod {\Bigl \{}e^{i\left(\alpha +{\frac {r\pi }{m}}\right)}\left({\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}\right)\\&+e^{-i\left(\alpha +{\frac {r\pi }{m}}\right)}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right){\Bigr \}}.\end{aligned}}$$

Let $\xi =x+iy,\eta =x-iy$, the above product becomes

$$\prod _{0}^{m-1}\left\{e^{i\left(\alpha +{\frac {r\pi }{m}}\right)}{\frac {\partial }{\partial \xi }}+e^{-i\left(\alpha +{\frac {r\pi }{m}}\right)}{\frac {\partial }{\partial \eta }}\right\}$$

which is equal to

$e^{(m-1){\frac {i\pi }{2}}}\prod \left\{e^{mi\alpha }\left({\frac {\partial }{\partial \xi }}\right)^{m}-e^{-mi\alpha }\left({\frac {\partial }{\partial \eta }}\right)^{m}\right\}$; when $\alpha =0,{\frac {\pi }{2m}}$,

this becomes

$e^{(m-1){\frac {i\pi }{2}}}\left\{\left({\frac {\partial }{\partial \xi }}\right)^{m}-(-1)^{m}\left({\frac {\partial }{\partial \eta }}\right)^{m}\right\}$ and

$e^{(m-1){\frac {i\pi }{2}}}\left[\left({\frac {\partial }{\partial \xi }}\right)^{m}+(-1)^{m}\left({\frac {\partial }{\partial \eta }}\right)^{m}\right]$.

From (7), we find

$${\begin{aligned}{\frac {\partial ^{n-m}}{\partial z^{n-m}}}\left({\frac {\partial }{\partial x}}\pm i{\frac {\partial }{\partial y}}\right)^{m}{\frac {1}{r}}&={\frac {(2n)!}{2^{n}n!}}{\frac {1}{r^{2n+1}}}\left[1-{\frac {r^{2}\Delta ^{2}}{2.2n-1}}+\dots \right]z^{n-m}(x\pm iy)^{m}\\&=(-1)^{n}{\frac {(2n)!}{2^{n}n!}}{\frac {1}{r^{n+1}}}(\cos m\phi \pm i\sin m\phi )\sin ^{m}\theta {\Bigl \{}\cos ^{n-m}\theta \\&-{\frac {(n-m)(n-m-1)}{2.2n-1}}\cos ^{n-m-2}+\dots \theta {\Bigr \}}\end{aligned}}$$

hence

$${\frac {\partial ^{n-m}}{\partial z^{n-m}}}\left({\frac {\partial }{\partial x}}\pm i{\frac {\partial }{\partial y}}\right)^{m}{\frac {1}{r}}=(-1)^{n}{\frac {(n-m)!}{r^{n+1}}}(\cos m\phi \pm i\sin m\phi )P_{n}^{m}(\cos \theta )$$

as we see on referring to (4); we thus obtain the formulae

$${\frac {\partial ^{n-m}}{\partial z^{n-m}}}\left\{\left({\frac {\partial }{\partial \xi }}\right)^{m}+\left({\frac {\partial }{\partial \eta }}\right)^{m}\right\}{\frac {1}{r}}=(-1)^{n}{\frac {(n-m)!}{2^{m-1}r^{n+1}}}\cos m\phi P_{n}^{m}(\cos \theta )$$

$${\frac {\partial ^{n-m}}{\partial z^{n-m}}}\left\{\left({\frac {\partial }{\partial \xi }}\right)^{m}-\left({\frac {\partial }{\partial \eta }}\right)^{m}\right\}{\frac {1}{r}}=(-1)^{n}{\frac {(n-m)!}{2^{m-1}r^{n+1}}}\sin m\phi P_{n}^{m}(\cos \theta )$$

It is thus seen that the tesseral harmonics of degree n and order m are those which have $n-m$ axes coincident with the z axis, and the other m axis distributed in the equatorial plane, at angular intervals $\pi /m$. The sectorial harmonics have all their axes in the equatorial plane.

8. Determination of the Poles of a given Harmonic.—It has been shown that a spherical harmonic $Y_{n}(x,y,z)$ can be generated by means of an operator

$f_{n}\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)$ acting upon ${\frac {1}{r}}$,

the function $f_{n}$ being so chosen that

$$Y_{n}(x,y,z)=(-1)^{n}{\frac {(2n)!}{2^{n}n!}}\left\{1-{\frac {r^{2}\nabla ^{2}}{2.2n-1}}+\dots \right\}f_{n}(x,y,z);$$

this relation shows that if an expression of the form

$(x^{2}+y^{2}+^{2})f_{n-2}(x,y,z)$

is added to $f_{n}(x,y,z)$, the harmonic $Y_{n}(x,y,z)$ is unaltered; thus if $Y_{n}$ be regarded as given, $f_{n}(x,y,z)=0$, is not uniquely determined, but has an indefinite number of values differing by multiples of $x^{2}+y^{2}+z^{2}$. In order to determine the poles of a given harmonic, $f_{n}$ must be so chosen that it is resolvable into linear factors; it will be shown that this can be done in one, and only one, way, so that the poles are all real.

If x, y, z are such as to satisfy the two equations $Y_{n}(x,y,z)=0$, $x^{2}+y^{2}+z^{2}=0$, the equation $f_{n}(x,y,z)$ is also satisfied; the problem of determining the poles is therefore equivalent to the algebraical one of reducing $Y_{n}$ to the product of linear factors by means of the relation $x^{2}+y^{2}+z^{2}=0$, between the variables. Suppose

$$Y_{n}(x,y,z)=A\prod _{s=1}^{m}(l_{s}x+m_{s}y+n_{s}z)+(x^{2}+y^{2}+z^{2})V_{n-2}(x,y,z),$$

we see that the plane $l_{s}x+m_{s}y+n_{s}z=0$ passes through two of the 2n generating lines of the imaginary cone $x^{2}+y^{2}+z^{2}=0,$ in which that cone is intersected by the cone $Y_{n}(x,y,z)=0$. Thus a pole $(l_{s},m_{s},n_{s})$ is the pole with respect to the cone $x^{2}+y^{2}+z^{2}=0$, of a plane passing through two of the generating lines; the number of systems of poles is therefore $n(2n-1)$, the number of ways of taking the 2n generating lines in pairs. Of these systems of poles, however, only one is real, viz. that in which the lines in each pair correspond to conjugate complex roots of the equations $Y_{n}=0$, $x^{2}+y^{2}+z^{2}=0$. Suppose

$${\frac {x}{\alpha _{1}+i\beta _{1}}}={\frac {y}{\alpha _{2}+i\beta _{2}}}={\frac {z}{\alpha _{2}+i\beta _{2}}}$$

gives one generating line, then the conjugate one is given by

$${\frac {x}{\alpha _{1}-i\beta _{1}}}={\frac {y}{\alpha _{2}-i\beta _{2}}}={\frac {z}{\alpha _{2}-i\beta _{2}}}$$

and the corresponding factor $lx+my+nz$ is

$${\begin{vmatrix}x&y&z\\\alpha _{1}+i\beta _{1}&\alpha _{2}+i\beta _{2}&\alpha _{3}+i\beta _{3}\\\alpha _{1}-i\beta _{1}&\alpha _{2}-i\beta _{2}&\alpha _{3}-i\beta _{3}\\\end{vmatrix}}$$

which is real. It is obvious that if any non-conjugate pair of roots is taken, the corresponding factor, and therefore the pole, is imaginary. There is therefore only one system of real poles of a given harmonic, and its determination requires the solution of an equation of degree 2n. This, theorem is due to Sylvester (Phil. Mag. (1876), 5th series, vol. ii., "A Note on Spherical Harmonics").

9. Expression for the Zonal Harmonic with any Axis.—The zonal surface harmonic, whose axis is in the direction

${\frac {x'}{r'}},{\frac {y'}{r'}},{\frac {z'}{r'}}$, is $P_{n}\left({\frac {xx'+yy'+zz'}{rr'}}\right)$

or $P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos \phi -\phi ')$; this is expressible as a linear function of the system of zonal, tesseral, and sectorial harmonics already found. It will be observed that it is symmetrical with respect to $(x,y,z)$ and $(x',y',z')$, and must thus be capable of being expressed in the form

$$a_{0}P_{n}(\cos \theta )P_{n}(\cos \theta ')+\sum _{1}^{n}a_{m}P_{n}^{m}(\cos \theta )P_{n}^{m}(\cos \theta ')\cos m(\phi -\phi ')$$

and it only remains to determine the co-efficients $a_{0},a_{1},\dots ,a_{m},\dots a_{n}$ To find this expression, we transform $(x'x+y'y+z'z)^{n}$, where x, y, z satisfy the condition $x^{2}+y^{2}+z^{2}=0$; writing $\xi =x+iy,\eta =x-iy,\xi '=x'+iy',\eta '=x'-iy'$, we have

$$(xx'+yy'+zz')^{m}=({\frac {1}{2}}\eta '\xi +{\frac {1}{2}}\xi '\eta +zz')^{n}$$

which equals

$$(xx'+yy'+zz')^{n}=(zz')^{n}+\sum \sum {\frac {n!}{a!b!(n-a-b)!}}\left\{{\frac {\eta '^{a}\xi '^{b}\eta ^{b}\xi ^{b}+\eta '^{b}\xi '^{a}\xi ^{b}\eta ^{a}}{2^{a+b}}}\right\}(zz')^{n-a-b}$$

the summation being taken for all values of a and b, such that $a+b\leq n$, $a>b$; the values $a=0,b=0$ corresponding to the term $(zz')^{n}$. Using the relation $\xi \eta =-z^{2}$, this becomes

$${\begin{aligned}(xx'+yy'+zz')^{n}=(zz')^{n}+\sum \sum &{\frac {(-1)^{b}}{2^{a+b}}}{\frac {n!}{a!b!(n-a-b)!}}(\eta '\xi ')^{b}z'^{n-a-b}\\&\{(\eta '\xi )^{a-b}+(\xi '\eta )^{a-b}\}z^{n-a+b},\end{aligned}}$$

putting $a-b=m$, the coefficient of $\xi ^{m}z^{n-m}$ on the right side is

$$\sum {\frac {(-1)^{b}}{2^{m+2b}}}{\frac {n!}{b!(m+b)!(n-m-2b)!}}(\xi '\eta ')^{b}\eta '^{m}z'^{n-m-2b},$$

from $b=0$ to $b={\frac {1}{2}}(n-m)$, or ${\frac {1}{2}}(n-m-1)$, according as $n-m$ is even or odd. This coefficient is equal to

$${\begin{aligned}&{\frac {n!}{2^{m}m!(n-m)!}}(x'-iy')^{m}{\Bigl \{}z'^{n-m}-{\frac {(n-m)(n-m-1)}{2.2m+2}}z'^{n-m-2}(x'^{2}+y'^{2})\\&+{\frac {(n-m)(n-m-1)(n-m-2)(n-m-3)}{2.4.2m+2.2m+3}}z'^{n-m-4}(x'^{2}+y'^{2})^{2}-{\Bigr \}}\end{aligned}}$$

in order to evaluate this coefficient, put $z=1$, $x'=i\cos \alpha$, $y'=i\sin \alpha$, then this coefficient is that of $(i\cos \alpha +\sin \alpha )^{m}$, or of $i^{m}e^{-mi\alpha }$ in the expansion of $(z'+ix'\cos \alpha +iy'\sin \alpha )^{n}$ in powers of $e^{-i\alpha }$ and $e^{i\alpha }$, this has been already found, thus the coefficient is

$${\frac {n!}{(n+m)!}}e^{-im\phi '}P_{n}^{m}(\cos \theta ')r'^{n}.$$

Similarly the coefficient of $\eta ^{m}z^{n-m}$ is

$${\frac {n!}{(n+m!)}}e^{+im\phi '}P_{n}^{m}(\cos \theta ')r'^{n};$$

hence we have

$${\begin{aligned}{\frac {1}{r'^{n}}}(xx'+yy'+zz')^{n}=z^{n}P_{n}^{m}(\cos \theta ')+n!\sum _{1}^{n}&P_{n}^{m}(\cos \theta ')\{\cos m\phi '(\xi ^{m}+\eta ^{m})\\&+i\sin m\phi '(\eta ^{m}-\xi ^{m})\}{\frac {z^{n-m}}{(n+m!)}}\end{aligned}}$$

In this result, change x, y, z into

$${\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}$$

and let each side operate on $1/r$, then in virtue of (10), we have

$${\begin{aligned}&(rr')^{n}P_{n}\left({\frac {xx'+yy'+zz'}{rr'}}\right)=P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos({\overline {\phi -\phi '}}))\\&=P_{n}(\cos \theta )P_{n}(\cos \theta ')+2\sum {\frac {(n-m)!}{(n+m)!}}P_{n}^{m}(\cos \theta )P_{n}^{m}(\cos \theta ')\cos m(\phi -\phi ')\end{aligned}}$$

which is known as the addition theorem for the function $P_{n}$.

It has incidentally been proved that

$${\begin{aligned}P_{n}^{m}(\cos \theta )={\frac {(n+m!)}{2^{m}(n-m)!}}&\sin ^{m}\theta {\Bigl \{}\cos ^{n-m}\theta \\&-{\frac {(n-m)(n-m-1)}{2.2m+2}}\cos ^{n-m-2}\theta \sin ^{2}\theta +{\Bigr \}}\end{aligned}}$$

which is an expression for $P_{n}^{m}(\cos \theta )$ alternative to (4).

10. Legendre's Coefficients.—The reciprocal of the distance of a point $(r,\theta ,\phi )$ from a point on the z axis distant r' from the origin is

$$(r^{2}-2rr'\mu +r'^{2})^{-{\frac {1}{2}}}$$

which satisfies Laplace's equation, $\mu$ denoting $\cos \theta$. Writing this expression in the forms

$${\frac {1}{r'}}\left\{1-2{\frac {r}{r'}}\mu +{\frac {r^{2}}{r'^{2}}}\right\}^{-{\frac {1}{2}}},{\frac {1}{r}}\left\{1-2{\frac {r'}{r}}\mu +{\frac {r'^{2}}{r^{2}}}\right\}^{-{\frac {1}{2}}}$$

it is seen that when $r<r'$, the expression can be expanded in a convergent series of powers of $r/r'$, and when $r'<r$ in a convergent series of powers of $r'/r$. We have, when $h^{2}(2\mu -h)^{2}<1$

$${\begin{aligned}(1-2h\mu +h^{2})^{-{\frac {1}{2}}}=1+h(2\mu -h)&+{\frac {1.3}{2.4}}h^{2}(2\mu -h)^{2}+\dots \\&+{\frac {1.3.5\dots 2n-1}{2.4\dots 2n}}h^{n}(2\mu -h)^{n}\end{aligned}}$$

and since the series is absolutely convergent, it may be rearranged as a series of powers of h, the coefficient of $h^{n}$ is then found to be

$${\frac {1.3.5\dots 2n-1}{1.2.3\dots n}}\left\{\mu ^{n}-{\frac {n(n-1)}{2.2n-1}}\mu ^{n-2}+{\frac {n(n-1)(n-2)(n-3)}{2.4.2n-1.2n-3}}\mu ^{n-4}\right\}$$

this is the expression we have already denoted by $P_{n}(\mu )$; thus

$$(1-2h\mu +h^{2})^{-{\frac {1}{2}}}=P_{0}(\mu )+hP_{1}(\mu )+\dots +h^{n}P_{n}(\mu )+\dots$$

the function $P_{n}(\mu )$ may thus be defined as the coefficient of $h^{n}$ in this expansion, and from this point of view is called the Legendre's coefficient or Legendre's function of degree n, and is identical with the zonal harmonic. It may be shown that the expansion is valid for all real and complex values of h and $\mu$, such that mod. h is less than the smaller of the two numbers mod. $(\mu \pm {\sqrt {\mu ^{2}-1}})$. We now see that

$$(r^{2}-2rr'\mu +r'^{2})^{-{\frac {1}{2}}}$$

is expressible in the form

$$\sum _{0}^{\infty }{\frac {r^{n}}{r'^{n+1}}}P_{n}(\mu )$$

when $r<r'$, or

$$\sum _{0}^{\infty }{\frac {r'^{n}}{r^{n+1}}}P_{n}(\mu )$$

when $r'<r$; it follows that the two expressions $r^{n}P_{n}(\mu )$, $r^{-n-1}P_{n}(\mu )$ are solutions of Laplace's equation.

The values of the first few Legendre's coefficients are

$$P_{0}(\mu )=1,P_{1}(\mu )=\mu ,P_{2}(\mu )={\frac {1}{2}}(3\mu ^{2}-1),P_{3}(\mu )={\frac {1}{2}}(5\mu ^{3}-3\mu )$$ $$P_{4}(\mu )={\frac {1}{8}}(35\mu ^{4}-30\mu ^{2}+3),P_{5}(\mu )={\frac {1}{8}}(63\mu ^{5}-70\mu ^{3}+15\mu )$$ $$P_{6}(\mu )={\frac {1}{16}}(231\mu ^{6}-315\mu ^{4}+105\mu ^{2}-5),P_{7}(\mu )={\frac {1}{16}}(429\mu ^{7}-693\mu ^{5}+315\mu ^{3}-35\mu ).$$

We find also

$P_{n}(1)=1,P_{n}(-1)=(-1)^{n}$

$P_{n}(0)=0$, or $(-1)^{{\frac {1}{2}}n}{\frac {1.3.5\dots n-1}{2.4\dots n}}$

according as n is odd or even; these values may be at once obtained from the expansion (13), by putting $\mu =1,0,-1$.

11. Additional Expressions for Legendre's Coefficients.—The expression (3) for $P_{n}(\mu )$ may be written in the form

$$P_{n}(\mu )={\frac {(2n)!}{2^{n}n!n!}}\mu ^{n}F\left(-{\frac {n}{2}},{\frac {1-n}{2}},{\frac {1}{2}}-n,{\frac {1}{\mu ^{2}}}\right)$$

with the usual notation for hypergeometric series. On writing this series in the reverse order

$$P_{n}(\mu )=(-1)^{{\frac {1}{2}}n}{\frac {n!}{2^{n}\left({\frac {1}{2}}n\right)!\left({\frac {1}{2}}n\right)!}}F\left(-{\frac {n}{2}},{\frac {n+1}{2}},{\frac {1}{2}},\mu ^{2}\right)$$

or

$$(-1)^{\frac {n-1}{2}}{\frac {n!}{{\frac {2^{n-1}n-1}{2}}!{\frac {n-1}{2}}!}}\mu F\left(-{\frac {n-1}{2}},{\frac {n}{2}}+1,{\frac {3}{2}},\mu ^{2}\right)$$

according as n is even or odd.

From the identity

$$(1-2h\cos \theta +h^{2})^{-{\frac {1}{2}}}=(1-he^{i\theta })^{-{\frac {1}{2}}}(1-he^{-i\theta })^{-{\frac {1}{2}}}$$

it can be shown that

$${\begin{aligned}P_{n}(\cos \theta )={\frac {1.3.5\dots 2n-1}{2.4.6\dots 2n}}&{\Bigl \{}\cos n\theta +{\frac {1.n}{1.2n-1}}\cos(n-2)\theta \\&+{\frac {1.3.n(n-1)}{1.2.(2n-1)(2n-3)}}\cos(n-4)\theta +{\Bigr \}}\end{aligned}}$$

By (13), or by the formula

$$P_{n}(\mu )={\frac {1}{2^{n}n!}}{\frac {d^{n}}{d\mu ^{n}}}(\mu ^{2}-1)^{n}$$

which is known as Rodrigue's formula, we may prove that

$${\begin{aligned}P_{n}(\cos \theta )=1-{\frac {(n+1)n}{1^{2}}}\sin ^{2}{\frac {\theta }{2}}&+{\frac {(n+2)(n+1)n(n-1)}{1^{2}.2^{2}}}\sin ^{4}{\frac {\theta }{2}}\dots \\&=F\left(n+1,-n,1,\sin ^{2}{\frac {\theta }{2}}\right)\end{aligned}}$$

Also that

$${\begin{aligned}P_{n}(\cos \theta )=\cos ^{2n}{\frac {\theta }{2}}&\left\{1-{\frac {n^{2}}{1^{2}}}\tan ^{2}{\frac {\theta }{2}}+{\frac {n^{2}(n-1)^{2}}{1^{2}.2^{2}}}\tan ^{4}{\frac {\theta }{2}}-\dots \right\}\\&=\cos ^{2n}{\frac {\theta }{2}}F\left(-n,-n,1,-\tan ^{2}{\frac {\theta }{2}}\right).\end{aligned}}$$

By means of the identity

$$(1-2h\mu +h^{2})^{-{\frac {1}{2}}}=(1-h\mu )^{-1}\left\{1+{\frac {h^{2}(1-\mu ^{2})}{(1-h\mu )^{2}}}\right\}^{-{\frac {1}{2}}}$$

it may be shown that

$${\begin{aligned}P_{n}(\cos \theta )=\cos ^{n}\theta &\left\{1-{\frac {n(n-1)}{2^{2}}}\tan ^{2}\theta +{\frac {n(n-1)(n-2)(n-3)}{2^{2}.4^{2}}}\tan ^{4}\theta -\dots \right\}\\&=\cos ^{n}\theta F\left(-{\frac {1}{2}}n,{\frac {1}{2}}-{\frac {1}{2}}n,1,-\tan ^{2}\theta \right).\end{aligned}}$$

Laplace's definite integral expression (6) may be transformed into the expression

$${\frac {1}{\pi }}\int _{0}^{\pi }{\frac {d\phi }{(\mu -{\sqrt {\mu ^{2}-1}}\cos \psi )^{n+1}}},$$

by means of the relation

$$(\mu +{\sqrt {\mu ^{2}-1}}\cos \phi )(\mu -{\sqrt {\mu ^{2}-1}}\cos \psi )=1.$$

Two definite integral expressions for $P_{n}(\mu )$ given by Dirichlet have been put by Mehler into the forms

$$P_{n}(\cos \theta )={\frac {2}{\pi }}\int _{0}^{\theta }{\frac {\cos \left(n+{\frac {1}{2}}\right)\phi }{\sqrt {2\cos \phi -2\cos \theta }}}d\phi ={\frac {2}{\pi }}\int _{\theta }^{\pi }{\frac {\sin \left(n+{\frac {1}{2}}\right)\phi }{\sqrt {2\cos \theta -2\cos \phi }}}d\phi .$$

When n is large, and $\theta$ is not nearly equal to 0 or to $\pi$, an approximate value of $P_{n}(\cos \theta )$ is $\{2/n\pi \sin \theta \}^{\frac {1}{2}}\sin \left\{(n+{\frac {1}{2}})\theta +{\frac {1}{4}}\pi \right\}$.

12. Relations between successive Legendre's Coefficients and their Derivatives.—If $(1-2h\mu +h^{2})^{-{\frac {1}{2}}}$ be denoted by u, we find

$$(1-2h\mu +h^{2}){\frac {\partial u}{\partial h}}+(h-\mu )u=0;$$

on substituting $\sum h^{n}P_{n}$ for u, and equating to zero the coefficient of $h^{n}$, we obtain the relation

$$nP_{n}-(2n-1)\mu P_{n-1}+(n-1)P_{n-2}=0.$$

From Laplace's definite integral, or otherwise, we find

$$(\mu ^{2}-1){\frac {dP_{n}}{d\mu }}=n(\mu P_{n}-P_{n-1})=-(n+1)(\mu P_{n}-P_{n+1}).$$

We may also show that

$$\mu {\frac {dP_{n}}{d\mu }}-{\frac {dP_{n-1}}{d\mu }}=nP_{n}$$ $$(n+1)P_{n}=-\mu {\frac {dP_{n}}{d\mu }}+{\frac {dP_{n+1}}{d\mu }}$$ $$(2n+1)P_{n}={\frac {dP_{n+1}}{d\mu }}-{\frac {dP_{n-1}}{d\mu }}$$ $$(2n+1)\mu {\frac {dP_{n}}{d\mu }}=(n+1){\frac {dP_{n-1}}{d\mu }}+n{\frac {dP_{n+1}}{d\mu }}$$ $$(2n+1)(\mu ^{2}-1){\frac {dP_{n}}{d\mu }}=n(n+1)(P_{n+1}-P_{n-1})$$ $${\frac {dP_{n}}{d\mu }}=(2n-1)P_{n-1}+(2n-5)P_{n-3}+(2n-9)P_{n-5}+\dots$$

the last term being $3P_{1}$ or $P_{0}$ according as n is even or odd.

13. Integral Properties of Legendre's Coefficients.—It may be shown that if $P_{n}(\mu )$ be multiplied by any one of the numbers 1, $\mu$, $\mu ^{2}$, ... $\mu ^{n-1}$ and the product be integrated between the limits 1, -1 with respect to $\mu$, the result is zero, thus

$$\int _{-1}^{1}\mu ^{k}P_{n}(\mu )d\mu =0,\alpha =0,1,2,\dots ,n-1.$$

To prove this theorem we have

$$\int _{-1}^{1}\mu ^{k}P_{n}(\mu )d\mu ={\frac {1}{2^{n}n!}}\int _{-1}^{1}\mu ^{k}{\frac {d^{n}}{d\mu ^{n}}}(\mu ^{2}-1)^{n}d\mu ,$$

on integrating the expression k times by parts, and remembering that $(\mu ^{2}-1)^{n}$ and its first $n-1$ derivatives all vanish when $\mu =\pm 1$, the theorem is established. This theorem derives additional importance from the fact that it may be shown that $AP_{n}(\mu )$ is the only rational integral function of degree n which has this property; from this arises the importance of the functions $P_{n}$ in the theory of quadratures.

The theorem which lies at the root of the applicability of the functions $P_{n}$ to potential problems is that if n and n' are unequal integers

$$\int _{-1}^{1}P_{n'}(\mu )P_{n}(\mu )d\mu =0,$$

which may be stated by saying that the integral of the product of two Legendre's coefficients of different degree taken over the whole of a spherical surface with its centre at the origin is zero; this is the fundamental harmonic property of the functions. It is immediately deducible from (18), for if $n'<n$, $P_{n'}(\mu )$ is a linear function of powers of $\mu$, whose indices are all less than n.

When $n'=n$ , the integral in (19) becomes $\int _{-1}^{1}\{P_{n}(\mu )\}^{2}d\mu$; to evaluate this we write it in the form

$${\frac {1}{2^{2n}n!n!}}\int _{-1}^{1}{\frac {d^{n}}{d\mu ^{n}}}(\mu ^{2}-1)^{n}{\frac {d^{n}}{d\mu ^{n}}}(\mu ^{2}-1)^{n}d\mu ;$$

on integrating n times by parts, this becomes

${\frac {(-1)^{n}}{2^{2n}n!n!}}\int _{-1}^{1}(\mu ^{2}-1)^{n}{\frac {d^{2n}}{d\mu ^{2n}}}(\mu ^{2}-1)^{n}d\mu ,$ or ${\frac {(2n)!}{2^{2n}n!n!}}\int _{-1}^{1}(1-\mu ^{2})^{n}\mu ,$

which on putting

$u={\frac {1}{2}}(1-\mu$, becomes $2{\frac {(2n)!}{n!n!}}\int _{-1}^{1}u^{n}(1-u)^{n}du,$

hence

$$\int _{-1}^{1}\{P_{n}(\mu )\}^{2}d\mu ={\frac {2}{2n+1}}.$$

14. Expansion of Functions in Series of Legendre's Coefficients.—If it be assumed that a function $f(\mu )$ given arbitrarily in the interval $\mu =-1$ to $+1$, can be represented by a series of Legendre's coefficients $a_{0}+a_{1}P_{1}(\mu )+a_{2}P_{2}(\mu )+\dots +a_{n}P_{n}(\mu )+\dots$ and it be assumed that the series converges in general uniformly within the interval, the coefficient a can be determined by using (19) and (20); we see that the theorem (19) plays the same part as the property $\int _{0}^{\pi }{\begin{matrix}\cos \\\sin \end{matrix}}n\theta {\begin{matrix}\cos \\\sin \end{matrix}}n'\theta d\theta =0$, $(n\neq n')$ does in the theory of the expansion of functions in series of circular functions. On multiplying the series by $P_{n}(\mu )$, we have

$$a_{n}\int _{-1}^{1}\{P_{n}(\mu )\}^{2}d\mu =\int _{-1}^{1}f(\mu )P_{n}(\mu )d\mu$$

hence

$$a_{n}={\frac {2n+1}{2}}\int _{-1}^{1}f(\mu )P_{n}(\mu )d\mu ,$$

hence the series by which $f(\mu )$ is in general represented in the interval is

$$\sum {\frac {2n+1}{2}}P_{n}(\mu )\int _{-1}^{1}f(\mu ')P_{n}(\mu ')d\mu '.$$

The proof of the possibility of this representation, including the investigation of sufficient conditions as to the nature of the function $f(\mu )$, that the series may in general converge to the value of the function requires an investigation, for which we have not space, similar in character to the corresponding investigations for series of circular functions (see Fourier's Series). A complete investigation of this matter is given by Hobson, Proc. Lond. Math. Soc., 2nd series, vol. 6, p. 388, and vol. 7, p. 24. See also Dini's Serie di Fourier.

The expansion may be applied to the determination at an external and an internal point of the potential due to a distribution of matter of surface density $f(\mu )$ placed on a spherical surface $r=a$. If

$V_{1}=\sum A'_{n}{\frac {r^{n}}{a^{n+!}}}P_{n}(\mu ),V_{0}=\sum A_{n}{\frac {a^{n}}{r^{n+1}}}P_{n}(\mu ),$

we see that $V_{1}$, $V_{0}$ have the characteristic properties of potential functions for the spaces internal to, and external to, the spherical surface respectively; moreover, the condition that $V_{1}$ is continuous with $V_{0}$ at the surface $r=a$ is satisfied. The density of a surface distribution which produces these potentials is in accordance with a known theorem in the potential theory, given by

$$\sigma ={\frac {1}{4\pi }}\left({\frac {\partial V_{1}}{\partial r}}-{\frac {\partial V_{0}}{\partial r}}\right)r=a,$$

hence

$\sigma ={\frac {1}{4\pi a^{2}}}\sum (2n+1)A_{n}P_{n}(\mu )$; on comparing this with the series (21),

we have $A_{n}=2\pi a^{2}\int _{-1}^{1}f(\mu ')P_{n}(\mu ')d\mu '$,

hence

$${\begin{cases}V_{1}&=2\pi a\sum {\frac {r^{n}}{a^{n}}}P_{n}(\mu )\int _{-1}^{1}f(\mu ')P_{n}(\mu ')d\mu '\\V_{0}&=2\pi a\sum {\frac {a^{n+1}}{r^{n+1}}}P_{n}(\mu )\int _{-1}^{1}f(\mu ')P_{n}(\mu ')d\mu '\\\end{cases}}$$

are the required expressions for the internal and external potentials due to the distribution of surface density $f(\mu )$.

15. Integral Properties of Spherical Harmonics.—The fundamental harmonic property of spherical harmonics, of which property (19) is a particular case, is that if $Y_{n}(x,y,z)$, $Z_{n'}(x,y,z)$ be two (ordinary) spherical harmonics, then,

$$\iint Y_{n}(x,y,z)Z_{n'}(x,y,z)dS=0,$$

when n and n' are unequal, the integration being taken for every element dS of a spherical surface, of which the origin is the centre. Since $\nabla ^{2}Y_{n}=0$, $\nabla ^{2}Z_{n'}=0$, we have

$$\iiint (Y_{n}\nabla ^{2}Z_{n'}-Z_{n'}\nabla ^{2}Y_{n})dxdydz=0,$$

the integration being taken through the volume of the sphere of radius r; this volume integral may be written

$${\begin{aligned}\iiint {\Bigl \{}{\frac {\partial }{\partial x}}\left(Y_{n}{\frac {\partial Z_{n'}}{\partial x}}-Z_{n'}{\frac {\partial Y_{n}}{\partial x}}\right)&+{\frac {\partial }{\partial y}}\left(Y_{n}{\frac {\partial Z_{n'}}{\partial y}}-Z_{n'}{\frac {\partial Y_{n}}{\partial y}}\right)\\&+{\frac {\partial }{\partial z}}\left(Y_{n}{\frac {\partial Z_{n'}}{\partial z}}-Z_{n'}{\frac {\partial Y_{n}}{\partial z}}\right){\Bigr \}}=0;\end{aligned}}$$

by a well-known theorem in the integral calculus, the volume integral may be replaced by a surface integral over the spherical surface; we thus obtain

$${\begin{aligned}\iint {\Bigl \{}{\frac {x}{r}}\left(Y_{n}{\frac {\partial Z_{n'}}{\partial x}}-Z_{n'}{\frac {\partial Y_{n}}{\partial x}}\right)&+{\frac {y}{r}}\left(Y_{n}{\frac {\partial Z_{n'}}{\partial y}}-Z_{n'}{\frac {\partial Y_{n}}{\partial y}}\right)\\&+{\frac {z}{r}}\left(Y_{n}{\frac {\partial Z_{n'}}{\partial z}}-Z_{n'}{\frac {\partial Y_{n}}{\partial z}}\right){\Bigr \}}dS=0;\end{aligned}}$$

on using Euler's theorem for homogeneous functions, this becomes

$${\frac {n'-n}{r}}\iint Y_{n}Z_{n'}dS=0,$$

whence the theorem (22), which is due to Laplace, is proved.

The integral over a spherical surface of the product of a spherical harmonic of degree n, and a zonal surface harmonic $P_{n}$ of the same degree, the pole of which is at $(x',y',z')$ is given by

$$\iint Y_{n}(x,y,z)P_{n}dS={\frac {4\pi }{2n+1}}a^{n+2}Y_{n}(x',y',z');$$

thus the value of the integral depends on the value of the spherical harmonic at the pole of the zonal harmonic. This theorem may also be written

$$\int _{0}^{2\pi }\int _{-1}^{1}V_{n}(\theta ,\phi )P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos {\overline {\phi -\phi '}})d\mu d\phi ={\frac {4\pi }{2n+1}}V_{n}(\theta ',\phi ').$$

To prove the theorem, we observe that $V_{n}$ is of the form

$$a_{0}P_{n}(\mu )+\sum _{1}^{n}(a_{m}\cos m\phi +b_{m}\sin m\phi )P_{n}^{m}(\mu );$$

to determine $a_{0}$ we observe that when $\mu =1$,

$$P_{n}(\mu )=1,P_{n}^{m}(\mu )=0;$$

hence $a_{0}$ is equal to the value $V_{n}(0)$ of $V_{n}(\theta ,\phi )$ at the pole $\theta =0$ of $P_{n}(\mu )$. Multiply by $P_{n}(\mu )$ and integrate over the surface of the sphere of radius unity, we then have

$${\begin{aligned}\int _{0}^{2\pi }\int _{-1}^{1}V_{n}(\theta ,\phi )P_{n}(\mu )d\mu d\phi &=a_{0}\int _{0}^{2\pi }\int _{-1}^{1}\{P_{n}(\mu )\}^{2}d\mu d\phi \\&={\frac {4\pi }{2n+1}}a_{0}={\frac {4\pi }{2n+1}}V_{n}(0),\end{aligned}}$$

if instead of taking $\mu =1$ as the pole of $P_{n}(\mu )$ we take any other point $(\mu ',\phi ')$ we obtain the theorem (23).

If $f(x,y,z)$ is a function which is finite and continuous throughout the interior of a sphere of radius R, it may be shown that

$${\begin{aligned}\iint Y_{n}(x,y,z)f(x,y,z)dS&=4\pi R^{2n+2}{\frac {2^{n}n!}{(2n+1)!}}{\Bigl \{}1+{\frac {R^{2}\nabla ^{2}}{2.2n+3}}\\&+{\frac {R^{4}\nabla ^{4}}{2.4.2n+3.2n+5}}+\dots \}Y_{n}\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)f(x,y,z)\end{aligned}}$$

where x, y, z are put equal to zero after the operations have been performed, the integral being taken over the surface of the sphere of radius R (see Hobson, "On the Evaluation of a certain Surface Integral," Proc. Land. Math. Soc. vol. xxv.).

The following case of this theorem should be remarked: If $f_{n}(x,y,z)$ is homogeneous and of degree n

$$\iint Y_{n}(x,y,z)f_{n}(x,y,z)dS=4\pi R^{2n+2}{\frac {2^{n}n!}{(2n+1)!}}Y_{n}\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)f(x,y,z)$$

if $f_{n}(x,y,z)$ is a spherical harmonic, we obtain from this a theorem, due to Maxwell (Electricity, vol. i. ch. ix.),

$$\iint Y_{n}(x,y,z)f_{n}(x,y,z)dS={\frac {4\pi R^{2n+2}}{2n+1}}{\frac {1}{n!}}{\frac {\partial ^{n}}{\partial h_{1}\partial h_{2}\dots \partial h_{n}}}f_{n}(x,y,z)$$

where $h_{1}h_{2}\dots h_{n}$ are the axes of $Y_{n}$. Two harmonics of the same degree are said to be conjugate, when the surface integral of their product vanishes; if $Y_{n},Z_{n}$ are two such harmonics, the addition of conjugacy is

$$Y_{n}\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)Z_{n}(x,y,z)=0.$$

Lord Kelvin has shown how to express the conditions that $2n+1$ harmonics of degree n form a conjugate system (see B. A. Report, 1871).

16. Expansion of a Function in a Series of Spherical Harmonics.—It can be shown that under certain restrictions as to the nature of a function $F(\mu ,\phi )$ given arbitrarily over the surface of a sphere, the function can be represented by a series of spherical harmonics which converges in general uniformly. On this assumption we see that the terms of the series can be found by the use of the theorems (22), (23). Let $F(\mu ,\phi )$ be represented by

$$V_{0}(\mu ,\phi )+V_{1}(\mu ,\phi )+\dots +V_{n}(\mu ,\phi )+\dots ;$$

change $\mu ,\phi$ into $\mu ',\phi '$ and multiply by

$$P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos {\overline {\phi -\phi '}}),$$

we have then

$${\begin{aligned}&\int _{0}^{2\pi }\int _{-1}^{1}F(\mu ',\phi ')P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos {\overline {\phi -\phi '}})d\mu 'd\phi '\\&=\int _{0}^{2\pi }\int _{-1}^{1}V_{n}(\mu ',\phi ')P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos {\overline {\phi -\phi '}})d\mu 'd\phi '\\&={\frac {4\pi }{2n+1}}V_{n}(\theta ,\phi )\end{aligned}}$$

hence the series which represents $F(\mu ,\phi )$ is

$${\frac {1}{4}}\pi \sum _{0}^{\infty }(2n+1)\int _{0}^{2\pi }\int _{-1}^{1}F(\mu ',\phi ')P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos {\overline {\phi -\phi '}})d\mu 'd\phi '$$

A rational integral function of $\sin \theta \cos \phi$, $\sin \theta \sin \phi$, $\cos \phi$ of degree n may be expressed as the sum of a series of spherical harmonics, by assuming

$$f_{n}(x,y,z)=Y_{n}+r^{2}Y_{n-2}+r^{4}Y_{n-4}+\dots$$

and determining the solid harmonics $Y_{n}$, $Y_{n-2}$, . . . and then letting $r=1$, in the result.

Since $\nabla ^{2}(r^{2s}Y_{n-2s})=2s(2n-2s+1)r^{2s-2}Y_{n-2s}$, we have

$$\nabla ^{2}f_{n}=2(2n-1)Y_{n-2}+4(2n-3)r^{2}Y_{n-4}+6(2n-5)r^{4}Y_{n-6}+\dots$$ $$\nabla ^{4}f_{n}=2.4(2n-3)(2n-5)Y_{n-4}+4.6(2n-5)(2n-7)r^{2}Y_{n-6}+\dots$$

the last equation being

$\nabla ^{n}f_{n}=n(n+1)(n-2)(n-1)\dots Y_{0}$, if n is even,

or

$\nabla ^{n-1}f_{n}=(n-1)(n+2)(n-3)n\dots Y_{1}$, if n is odd

from the last equation $Y_{0}$ or $Y_{1}$ is determined, then from the preceding one $Y_{2}$ or $Y_{3}$, and so on. This method is due to Gauss (see Collected Works, v. 630).

As an example of the use of spherical harmonics in the potential theory, suppose it required to calculate at an external point, the potential of a nearly spherical body bounded by $r=a(1+\epsilon u)$, the body being made of homogeneous material of density unity, and u being a given function of $\theta$, $\phi$, the quantity $\epsilon$ being so small that its square may be neglected. The potential is given by

$$\int _{0}^{2\pi }\int _{-1}^{1}\int _{0}^{a(1+\epsilon u')}\{r^{2}+r'^{2}-2rr'\cos \gamma \}^{-{\frac {1}{2}}}dr'd\mu 'd\phi ',$$

where $\gamma$ is the angle between r and r'; now let u' be expanded in a series

$$Y_{0}(\mu ',\phi ')+Y_{1}(\mu ',\phi ')+\dots +Y_{n}(\mu ',\phi ')+\dots$$

of surface harmonics; we may write the expression for the potential

$${\begin{aligned}\int _{0}^{2\pi }\int _{-1}^{1}\int _{0}^{a(1+\epsilon u')}&{\Bigl \{}{\frac {1}{r}}+{\frac {r'}{r^{2}}}P_{1}(\cos \gamma )+\dots \\&+{\frac {r'^{n}}{r^{n+1}}}P_{n}(\cos \gamma )+\dots {\Bigr \}}r'^{2}dr'd\mu 'd\phi '\end{aligned}}$$

which is,

$${\begin{aligned}\int _{0}^{2\pi }\int _{-1}^{1}{\Bigl \{}{\frac {a^{3}}{3r}}(1+3\epsilon u')&+{\frac {1}{4}}{\frac {a^{4}}{r^{2}}}(1+4\epsilon u')P_{1}+\dots \\&+{\frac {1}{n+3}}{\frac {a^{n+3}}{r^{n+1}}}(1+{\overline {n+3}}\epsilon u')P_{n}(\cos \gamma ){\Bigr \}}d\mu 'd\phi '\end{aligned}}$$

on substituting for u' the series of harmonics, and using (22), (23), this becomes

$${\begin{aligned}4\pi a^{2}{\Bigl [}{\frac {1}{3}}{\frac {a}{r}}+\epsilon {\Bigl \{}{\frac {a^{3}}{3r^{2}}}Y_{1}(\mu ,\phi )&+{\frac {a^{3}}{5r^{3}}}Y_{2}(\mu ,\phi )+\dots \\&+{\frac {a^{n+1}}{(2n+1)r^{n-1}}}Y_{n}(\mu ,\phi )+\dots {\Bigr \}}{\Bigr ]}\end{aligned}}$$

which is the required potential at the external point $(r,\theta ,\phi )$.

17. The Normal Solutions of Laplace's Equation in Polars.—If $h_{1}$, $h_{2}$, $h_{3}$ be the parameters of three orthogonal sets of surfaces, the length of an elementary arc ds may be expressed by an equation of

the form $ds^{2}={\frac {1}{H_{1}^{2}}}dh_{1}^{2}+{\frac {1}{H_{2}^{2}}}dh_{2}^{2}+{\frac {1}{H_{3}^{2}}}dh_{3}^{2}$, where $H_{1}$, $H_{2}$, $H_{3}$ are functions of $h_{1}$, $h_{2}$, $h_{3}$, which depend on the form of these parameters; it is known that Laplace's equation when expressed with $h_{1}$, $h_{2}$, $h_{3}$ as independent variables, takes the form

$${\frac {\partial }{\partial h_{1}}}\left({\frac {H_{1}}{H_{2}H_{3}}}{\frac {\partial V}{\partial h_{1}}}\right)+{\frac {\partial }{\partial h_{2}}}\left({\frac {H_{2}}{H_{3}H_{1}}}{\frac {\partial V}{\partial h_{2}}}\right)+{\frac {\partial }{\partial h_{3}}}\left({\frac {H_{3}}{H_{1}H_{2}}}{\frac {\partial V}{\partial h_{3}}}\right)=0.$$

In case the orthogonal surfaces are concentric spheres, co-axial, circular cones, and planes through the axes of the cones, the parameters are the usual polar co-ordinates r, $\theta$, $\phi$, and in this case $H_{1}=1,H_{2}={\frac {1}{r}},H_{3}={\frac {1}{r\sin \theta }}$, thus Laplace's equation becomes

$${\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial V}{\partial r}}\right)+{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial V}{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}V}{\partial \phi ^{2}}}=0.$$

Assume that $V=R\Theta \Phi$ is a solution, R being a function of r only, $\Theta$ of $\theta$ only, $\Phi$ of $\phi$ only; we then have

$${\frac {1}{R}}{\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)+{\frac {1}{\Theta \sin \theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {d\Theta }{d\theta }}\right)+{\frac {1}{\sin ^{2}\theta .\Phi }}{\frac {d^{2}\Phi }{d\phi ^{2}}}=0.$$

This can only be satisfied if ${\frac {1}{R}}{\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)$ is a constant, say $n(n+1)$, ${\frac {1}{\Phi }}{\frac {d^{2}\Phi }{\phi ^{2}}}$ is a constant, say $-m^{2}$, and $\Theta$ satisfies the equation

$${\frac {1}{\sin \theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {d\Theta }{d\theta }}\right)+\left\{n(n+1)-{\frac {m^{2}}{\sin ^{2}\theta }}\right\}\Theta =0,$$

if we write u for $\Theta$, and $\mu$ for $\sin \theta$, this equation becomes

$${\frac {d}{d\mu }}\left\{(1-\mu ^{2}){\frac {du}{d\mu }}\right\}+\left\{n(n+1)-{\frac {m^{2}}{1-\mu ^{2}}}\right\}u=0.$$

From the equations which determine R, $\Theta$, $u$, it appears that Laplace's equation is satisfied by

$${\begin{matrix}r^{n}\\r^{-n-1}\end{matrix}}{\begin{matrix}\cos \\\sin \end{matrix}}m\phi .u{\begin{matrix}m\\n\end{matrix}}$$

where u is any solution of (26); this product we may speak of as the normal solution of Laplace's equation in polar co-ordinates; it will be observed that the constants n, m may have any real or complex values.

18. Legendre's Equation.—If in the above normal solution we consider the case $m=0$, we see that

$${\begin{matrix}r^{n}\\r^{-n-1}\end{matrix}}u_{n}$$

is the normal form, where $u_{n}$ atisfies the equation

$${\frac {d}{d\mu }}\left\{(1-\mu ^{2}){\frac {du}{d\mu }}\right\}+n(n+1)u=0,$$

known as Legendre's equation; we shall here consider the special case in which n is a positive integer. One solution of (27) will be the Legendre's coefficient $P_{n}(\mu )$, and to find the complete primitive we must find another particular integral; in considering the forms of solution, we shall consider $\mu$ to be not necessarily real and between $\pm 1$. If we assume

$$u=\mu ^{m}+\alpha _{2}\mu ^{m-2}+\alpha _{4}\mu ^{m-4}+\dots$$

as a solution, and substitute in the equation (27), we find that $m=n$, or $-n-1$, and thus we have as solutions, on determining the ratios of the coefficients in the two cases,

$$\alpha \left\{\mu ^{n}-{\frac {n(n-1)}{2.2n-1}}\mu ^{n+2}+\dots \right\}$$

and

$$\beta \left\{{\frac {1}{\mu ^{n+1}}}+{\frac {(n+1)(n+2)}{2.2n+3}}{\frac {1}{\mu ^{n+3}}}+{\frac {(n+1)(n+2)(n+3)(n+4)}{2.4.2n+3.2n+5}}{\frac {1}{\mu ^{n+5}}}+\dots \right\}$$

the first of these series is (n integral) finite, and represents $P_{n}(\mu )$, the second is an infinite series which is convergent when ${\text{mod}}~\mu >1$. If we choose the constant $\beta$ to be ${\frac {1.2.3\dots n}{3.5\dots 2n+1}}$, the second solution may be denoted by $Q_{n}(\mu )$, and is called the Legendre's function of the second kind, thus

$${\begin{aligned}Q_{n}(\mu )={\frac {1.2.3\dots n}{3.5\dots 2n+1}}&\left\{{\frac {1}{\mu ^{n+1}}}+{\frac {(n+1)(n+2)}{2.2n+3}}{\frac {1}{\mu ^{n+3}}}+\dots \right\}\\&={\frac {1.2.3\dots n}{3.5\dots 2n+1}}{\frac {1}{\mu ^{n+1}}}F\left({\frac {n+1}{2}},{\frac {n+2}{2}},{\frac {2n+3}{2}},{\frac {1}{\mu ^{2}}}\right).\end{aligned}}$$

This function $Q_{n}(\mu )$, thus defined for ${\text{mod}}~\mu >1$, is of considerable importance in the potential theory. When ${\text{mod}}~\mu <1$, we may in a similar manner obtain two series in ascending powers of $\mu$, one of which represents $P_{n}(\mu )$, and a certain linear function of the two series represents the analytical continuation of $Q_{n}(\mu )$, as defined above. The complete primitive of Legendre's equation is

$$u=AP_{n}(\mu )+BQ_{n}(\mu ).$$

By the usual rule for obtaining the complete primitive of an ordinary differential equation of the second order when a particular integral is known, it can be shown that (27) is satisfied by

$$P_{n}(\mu )=\int ^{\mu }{\frac {d\mu }{(\mu ^{2}-1)\{P_{n}(\mu )\}^{2}}}$$