of order i. (rr')'Q, = p2i + 1 Σ Σ Σ

1.2... j. 1. 2 ..... k. 1.2...l d§'dŋ*dz' (§ŋ +2°) !

Of course we have in this case

go2 = {n + 22, g'2 = { 'n' + z'2,

...(56).

Biaxal har

monic expressed in symmetrical series of differential coefficients.

And, just as above, we see that this expression, obviously a homogeneous function of έ', n', z', of degree i, and also of 7, §, 2, involves these two systems of variables symmetrically.

of

Now, as we have seen above, all the ith differential coefficients

Hence r'Q, viewed as a function of z, έ, 7, is expressed by these 2+1 terms, each with a coefficient involving z', §, n'. And because of the symmetry we see that this coefficient must be the same function of ', ', έ', into some factor involving none of these variables (~, έ, n), (≈', n', §'). Also, by the symmetry with reference to έ, n' and ŋ, έ', we see that the numerical factor must be the same for the terms similarly involving έ, n' on the one hand, and 7, έ' on the other. Hence,

The value of E is obtained thus:-Comparing the coefficient of the term (z)'-'(§n')' in the numerator of the expression which (56) becomes when the differential coefficient is expanded, with the coefficient of the same term in (57), we have

1

which is the same, the coefficient of 'n' in p′2+1

From this, with the value (42) for M, we find E as above.

(y) We are now ready to reduce the expansion of Q, to a real trigonometrical form. First, we have, by (33),

Let now

(¿n')' + (§'n)' = 2(rr' sin 0 sin 0')' cos s (pp)... (59).

pressed in symmetrical series of differential coefficients.

(s)
i

(that is to say, C=", in accordance with the previous notation,) and let the corresponding notation with accents apply to '. Then, by the aid of (57), (58), and (59), we have

harmonic.

of which, however, the first term (that for which s=0) must be surface halved.

(2) As a supplement to the fundamental proposition [S;S;dw=0, (16) of (g), and the corresponding propositions, (43) and (44), regarding elementary terms of harmonics, we are now prepared to evaluate ffSda.

First, using the general expression (37) investigated above for S, and modifying the arbitrary constants to suit our present Fundanotation, we have

mental definite integral investigated.

To evaluate the definite integral in the second member, we have
only to apply the general theorem (52) for expansion, in terms of
surface harmonics, to the particular case in which the arbitrary
function F(E) is itself the harmonic, cos Thus, remem-
bering (16), we have

(8)

Απ

0

Fundamental defi. nite integral evaluated.

Spherical harmonic synthesis of arbitrary

function

concluded.

Using here for Q, its trigonometrical expansion just investigated, and performing the integration for ' between the stated limits, we find that cos 84 may be divided out, and (omitting the accents in the residual definite integral) we conclude,

[" sin 0(5")"d0 =

0

1 3 2.

1.2...(i-8)

2i+1 (81) (28+1) (28+2)... (28+i−8)

This holds without exception for the case

2
2i + 1'

...(65).

8 = 0, in which

It is convenient here to

the second member becomes
recal equation (44), which, when expressed in terms of
instead of (m, n) 9 becomes

(66),

where i and i must be different. The properties expressed by these two equations, (65) and (66), may be verified by direct integration, from the explicit expression (60) for ; and to do so will be a good analytical exercise on the subject.

(a) Denote for brevity the second member of (65) by (i, 8), so that

θ, Φ

Suppose the co-ordinates 0, & to be used in (52); so that a,
are the three co-ordinates of P, and we may take dơ=a2 sin 0d0dò̟.
Working out by aid of (61), (65), the processes indicated
symbolically in (52), we find

of the

which is the explicit form most convenient for general use, expansion of an arbitrary function of the co-ordinates 6, in spherical surface harmonics. It is most easily proved, [when

harmonic

207 once the general theorem expressed by (66) and (65) has been in Spherical any way established,] by assuming the form of expansion (68), analysis of and then determining the coefficients by multiplying both mem- function. bers by cos 84 sin 0d0dp, and again by sin so sin də də, and integrating in each case over the whole spherical surface.

arbitrary

preceding

(b) In what precedes the expansions of surface harmonics, Review of whether complete or not, have been obtained solely by the differ- expansions

1

and investigations of

1 entiation of with reference to rectilineal rectangular co- properties. ordinates x, y, z. The expansions of the complete harmonics have been found simply as expressions for differential coefficients, or for linear functions of differential coefficients of The expansions of harmonics of fractional and imaginary orders have been inferred from the expansions of the complete harmonics merely by generalizing their algebraic forms. The properties of the harmonics have been investigated solely from the differential equation

in terms of the rectilineal rectangular co-ordinates. The original investigations of Laplace, on the other hand, were founded exclusively on the transformation of this equation into polar co-ordinates. In our first edition this transformation was not given-we now supply the omission, not only on account of the historical interest attached to "Laplace's equation" in terms of polar co-ordinates, but also because in this form it leads directly by the ordinary methods of treating differential equations, to every possible expansion of surface harmonics in polar co-ordinates.

(c) By App. Z (g)(14) we find for Laplace's equation (20) transformed to polar co-ordinates,

Laplace's equation for surface harmonic in polar coordinates.

which is the celebrated formula commonly known in England as "Laplace's Equation" for determining S, the "Laplace's coefficient" of order i; i being an integer, and the solutions admitted or sought for being restricted to rational integral functions of cos 0, sin ✪ cos & and sin ✪ sin 4.

(d') Doing away now with all such restrictions, suppose i to be any number, integral or fractional, real or imaginary, only if imaginary let it be such as to make i(i-1) real [compare § (o)] above. On the supposition that S, is a rational integral function of cos 0, sin cos & and sin @ sin ø, it would be the sum of

terms such as

sin

COS

sp. Now, allowing s to have any value

integral or fractional, real or imaginary, assume

This will be a form of particular solution adapted for application to problems such as those referred to in $$ (1), (m) above; and (73) gives, for the determination of (",

Definition of "Laplace's functions."

(e) When i and are both integers we know from §() above, and we shall verify presently, by regular treatment of it in its present form, that the differential equation (75) has for one solution a rational integral function of sin and cos 0. It is this solution that gives the "Laplace's Function," or the "complete surface harmonic" of the form ""

()sin

i COS

sp. But being a

differential equation of the second order, (75) must have another distinct solution, and from § (h) above it follows that this second solution cannot be a rational integral function of sin 0, cos 0. It may of course be found by quadratures from the rational integral solution according to the regular process for finding the second particular solution of a differential equation of the second order when one particular solution is known. Thus denoting by (*) any solution, as for example the known rational integral solution expressed by equation (38), or (36) or (40) above, or § 782 (e) or (f) with (5) below, we have for the complete