Page:EB1911 - Volume 25.djvu/673

$${\displaystyle P_{n}(\mu )={\frac {(2n)!}{2^{n}n!n!}}\left\{\mu ^{n}-{\frac {n(n-1)}{2.2n-1}}\mu ^{n-2}+{\frac {n(n-1)(n-2)(n-3)}{2.4.2n-1.2n-3}}\mu ^{n-4}\dots \right\}}$$

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

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

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

In the present notation we have

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

if we put ${\displaystyle \alpha =0}$, we thus have

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

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

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

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

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

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

denote any rational integral operator,

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

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 ${\displaystyle 1/r}$, 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.)

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

which is a particular case of the more general theorem

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

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

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

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 ${\displaystyle (l_{\iota }x+m_{\iota }y+n_{\iota }z)/r}$ of the angle between the radius vector r to a point ${\displaystyle (x,y,z)}$ and the axis ${\displaystyle (l_{\iota },m_{\iota },n_{\iota })}$ will be denoted by ${\displaystyle \lambda _{\iota }}$; the cosine of the angle between two axes is ${\displaystyle l_{\iota }l_{\gamma }+m_{\iota }m_{\gamma }+n_{\iota }n_{\gamma }}$, which will be noted by ${\displaystyle \mu _{\iota \gamma }}$. The operation

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

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

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

when

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

this limit is

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

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

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

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

where ${\displaystyle Y_{n}}$ 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 ${\displaystyle Y_{n}r^{n},Y_{n}r^{-n-1}}$. Any spherical harmonic is completely specified by means of its poles.

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

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

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

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

and generally

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

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

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

where S denotes a summation with respect to m from ${\displaystyle m=0}$ to ${\displaystyle m={\frac {1}{2}}n}$, or ${\displaystyle {\frac {1}{2}}(n-1)}$, 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:—

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

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

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

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

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