To take a very simple example: If a charge of electricity is placed on a spherical conductor of radius a, it is known that it will so distribute itself that all points on the surface will be at the same potential - .
a Now - M - M
(cos 0) and is a Surface Zonal a a Harmonic. Hence any point at the distance r
from the centre of the conductor is at potential M a ae Po (cos 8) or
r
If the value of V on the surface of the sphere had been less simple, say V = F(0) = f(cos 0) -=- Au), then f(p)would have had to be expressed in the form aiPi(p) a2P2(p) .. . before we could have used the simple method illustrated above. This can be done by the aid of the formula f(p)=a0Po(p) +aiPi(P) 4-02P2(11)+a,P,(p)+ . . . , where am=-- 2m2 1 ri f (x) (x)dx, the de- 2 1 velopment in question holding good when- I< /A < 1.
For instance, let one-half of the surface of a homogeneous sphere be kept at the tempera ture zero and the other half at the temperature 1; to find the stationary temperature u of any internal point. Here f(m) = 1, 0
2 Letting nr=0, 1, 2, . . . , successively, and using the corresponding values 1, x, (3x' - 1) etc., of we get = = 1, a, = 0, a, 0, . . . and f(p)=4 + p,(p) - Pa(P) ÷,11Pi(ii) - .. .
If a is the radius of the sphere, the required temperature 1 3 r 7 1 rs u= Pi(cos 0) . 11 1 3 - 12 . 2 - . 4 al - P, (cos 9). . . . Tables giving the numerical values of the Surface Zonal Harmonics have been computed and are accessible, and by their aid numerical results can be obtained in such problems as those we have been considering as readily as if we were using simple trigonometric functions. The following is such a table carried only to three places.
Legendrians were first used by the Legendre in a paper published in 1785 on the attraction of solids of revolution.
Laplace's Coefficients.- Pm(cos y), where cos y a: cos 8 cos el + sine, sin ei cos (p- el), and is the angle made by the radius vector with a fixed line through the origin whose direction is given by the angles 01 and #1, is called a La place's Coefficient or Laplacian, the fixed line .being called the Axis and its intersection with the unit sphere the Pole of the Laplacian. A Surface Zonal Harmonic 0) is merely a Laplacian whose axis coincides with the axis of co-ordinates. rinPm(cosy) and rif, 1-1 Pm(cos
are solutions of Laplace's Equation (III). The first is harmonic within and the second without the unit sphere.
Laplacians may be used in problems symmet rical about an axis if the axis does not coin cide with the axis of co-ordinates just as Zonal Harmonics are used when the problem is symmetrical about the polar axis.
Laplacians were first used by Laplace, in one of the most remarkable memoirs ever written, in determining the attraction of a Spheroid. The paper in question was published in 1782.
Spherical Harmonics.-A Surface Spherical Harmonic of the mth degree may be most simply defined as the function obtained by dividing a rational, integral, homogeneous, algebraic polynomial of the mth degree in x, y, a with satisfies Laplace's Equation (I), by 2,2 rm, that is, by (x' + y' zi)i. For example, I 1 - (x -Fs) , - (xi 1-xy - (2x4-3xyb--3xsi) r are Surface Spherical Harmonics of the first degree, of the second degree and of the third degree, respectively.
It is clear that res satisfies Laplace's Equation. The same thing can be shown of 1 Th - e first is harmonic within, the second r'"+ I'm without, the unit sphere. They are known as Solid Spherical Harmonics.
It is clear that if the value of V on the sur face of a sphere whose centre is the origin can be expressed as a sum of terms each of which is a surface Spherical Harmonic, its value at any point not on the surface is the sum of the appropriate corresponding Solid Spherical Har monics.
It can be shown by transforming from spherical to rectangular co-ordinates that the Surface Zonal Harmonic Pm(p) or 0) and the Laplacian Pm(cos y) are Surface Spherical Harmonics, and by the reverse trans formation that the general Surface Spherical Harmonic I'm can be formulated as Ym = A P m(p) 11-= P=01 • [(An Cos nli+Bn sin nifr) a= 1 A function given arbitrarily on the surface of the unit sphere, i.e., a function of 0 and • if expressed as a function of cos B and can be developed into a series of Surface Spherical Harmonics by the formulas xi=x, (P,O)= fil.o.P 1.W + Il(Anos cos a 0 ns=0 n=1 + Bn, m sin nqi) sinn 0 1, dpn J Ao,ii= 2m + f (P 0)1'1,101)0, 47r 0 --1 27* n) • tin, In 2m + 1 (m—n) ! dnPm(p) f(p sb) cos nO sine 0 f 27 +n)! Bn, 2nt +1 (m —ft)! , u, 0) sin n0 sins o .1 bThe following theorems concerning the inte gration of Spherical Harmonics are important. We give them without proof.