Page:EB1911 - Volume 25.djvu/673

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651
SPHERICAL HARMONICS


r * w 2"»!n!< M 2.2n-i M T 2.4.2W-1 .2n~3 M i w

p"(„) = (|!*1; — ^(1 -u')i m \ u*-"

» w 2 v n!(n-w)! v M ;i (^

(n-w)(w-OT-i) _ , ) . v

2.2n-i M -r- ■• j v<w

Every ordinary harmonic of degree n is expressible as a linear func- tion of the system of 2» + i zonal, tesseral and sectorial harmonics of degree «; thus the general form of the surface harmonic is

«  OoP„ (m) + 2 (on cos m<£ +6 m sin m$) P„ 00 • (5)

m = i In the present notation we have

P.U)+22i™-££-j-^P lt (Ai)cos»»(*-a) I

if we put = 0, we thus have

n

(cos 0+i sin cos <*>)" = P„(cos 0) +22 t '"' / i„\| P„(cos 0) cos »»#,

1 â–

m

from this we obtain expressions for P„(cos 0), P n (cos 0) as definite integrals

P„(cos 0) =- y (cos + i sin cos <t>) n d<t> j

iP., (cos 0) =- i" (cos + i sin cos #)"cos nt(t>d<t>. \

by Differentiation, — The


(6)


1 W+*n

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

f(-^^-)

J \dx dy dz]

denote any rational integral operator,


J \dx' W dz) v


is a solution of 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 i/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.)

L I J -._! _2n+l


' n \dx' dy' dz) r '


+ :


2"nl r" r 4 V 4


-!â– -


-...(/.<-


2 .\.2n — l 2n — 3 which is a particular case of the more general theorem

f (A, A, A) F(r) = \ 2»-^£ +HT. 2 f ~'F 7 * + }n \dx dy dz) t ( r >-} 2 d(r*)" + TT Jp)^ 1 + â– â–


y,z)


(7)


V-


d"-'F


ry+--. \fn(x,y,z)


(7 1 ),


of the singular point. Let a singular point of degree zero, and strength e , be on an axis hi, at a distance ao from the origin, and also suppose that the origin is a singular point of strength — e ; let eo be indefinitely increased, and 00 indefinitely diminished, but so that the product «o<>o is finite and equal to eo ! the origin is then said to be a singular point of the first degree, of strength e u the axis being hi. Such a singular point is frequently called a doublet. In a similar manner, by placing two singular points of degree, unity and strength, e u —ei, at a distance 01 along an axis h 2 , and at the origin respectively, when «i is indefinitely increased, and 01 diminished so that dai is finite and=e 2 , we obtain a singular point of degree 2, strength e% at the origin, the axes being hi, hi. Proceeding in this manner we arrive at the conception of a singular point of any degree n, of strength e„ at the origin, the singular point having any n given axes hi, }h,. . .h„. If e n -i 4>n-i (x, y, 2) is the potential due to a singular point at the origin, of degree n — 1, and strength e„_i, with axes hi, h*,—h*-i, the potential of a singular point of degree n, the new axis of which is h n , is the limit of


s\ d(r 2 )»"

where /„(*, y, z) is a rational integral homogeneous function of degree n. The harmonic of positive degree n corresponding to that of degree — n — I in the expression (7) is

} 2.2»-I T 24.2(l-1.2«-3 ) J X ' •" '

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 /, m, n drawn from the origin, is called an axis, and the point where this axis cuts the sphere is called the pole of the axis. Differ- ent axes will be denoted by suffixes attached to the direction-cosines ; the cosine (l,x-\-m,y+niz)jr of the angle between the radius vector r to a point (x, y, z) and the axis (h, m,, ni), will be denoted by Xi; the cosine of the angle between two axes is Uy+m,m y -\-nMy, which will be denoted by n,y. The operation


when

this limit is


a

dx dy dz


I'-zz+m^+nr


e„_i <£n-i (* — J»a, y— m n a, z—n„a) — e„_! 4>„_i (x, y, z) ; Lo = o, Le n _i = o>, Laen_i = e n ;


„ /; 5><£»-l_L«, d0„_i 1 „ 1


d<t>«-i 1 „ d<t>*-i \ -. ,

-dy- +nn ^r)' 0T _e "a


-<t>n-l-


dx dy ' "" dz J'"' - n dhJ

Since <j>e = i/r, we see that the potential V, due to a singular point at the origin of strength e,„ and axes hi, hi, . . .h n is given by

^=^ n ^hidZ..dh j (8)

6. Expression for a Harmonic with given Poles. — The result of performing the operations in (8) is that V» is of the form

where Y„ 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„r", Ynr - " -1 . Any spherical harmonic is completely specified by means of its poles.

In order to express Y„ in terms of the positions of its poles, we apply the theorem (7) to the evaluation of V„ in (8). On putting r=n f n {x, y, z) —Xl(l,x+mry+n r z), we have t =1 (2n!) I /. rV , r 4 V 4


performed upon any function of x, y, z, is spoken of as differentiation with respect to the axis (L, m>, «i), and is denoted by d/dh,. The potential function Vo = ft>/r is defined to be the potential due to a singular point of degree zero at the origin ; eo is called the strength


V _ (2"0 1 /. fV I " 2"»!n! r n \ 2.2n~i


2 .\.2n — 1 .in


=i-)


X


n (l,x +m r y+n,z).

1 By ZGi'X" -25 ) we shall denote the sum of the products of 5 of the quantities p, and n — 2s of the quantities X; in any term each suffix is to occur once, and once only, every possible order being taken. We find

Mlx+my+nz) =2(X")r\ A 2 n(te +my+nz) =22( 1 u 1 X' , - 2 )r"- 2 , and generally

A 2m n(lx+my+nz) =2 m m ! 2(^" , X"- 2 "')r n - :!m ;

thus we obtain the following expression for Y n , the surface har- monic which has given poles hi, hi, . . ,h„;

d" ±

r


Yn = r n+lL-Lt


n\ dhidhi. . .dh„


(-1)"


, (2«— 2m)! 2"~'"n\ (n—m)


,2(X»


V") \


(9)


m =


where S denotes a summation with respect to m from m = o to m = \n, or \(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: —

~ Yi = cos PA, Y s = J(3 cos PA cos PB-cos AB),

Y« = 4(J5 cos PA cos PB cos PC -cos PA cos BC-cos PB cos CA

--cos PC cos AB), Y 4 = K35 cos PA cos PB cos PC cos PD -52 cos PA cos PB cos CD

+ 2 cos AB cos CD), 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

( — i)"r» +1 d» I. n ! dz' r '