Page:EB1911 - Volume 25.djvu/677

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

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

the lower limit being arbitrary.