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Expansions of the biaxal harmonic,

of order i. (')'Q = p2 + 1 2

Biaxal harmonic expressed in symmetrical series of differential



Of course we have in this case



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

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+ ΣΕ





· k . 1 . 2 ..... l d§1dŋ*dz3 (§n + z2)2
1.2...j. 1.2...k.1.2...l d§'dŋ*dz'

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


are reducible to the 2i + 1 independent forms

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

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d\'1 d 1


dz' r' dz r


di 1


1 d' 1
\dz'-'d§'3 7' dz1 ́'dŋ'r" dz'-'dŋ''r' dzi¬d§3 r ̧



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1.2...8.1.2... (i-8).1.3... (8-1). (28+1)(28+2)...(i+8)





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

1.2... (-s).1.2... 8

E M2......... (58),


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where M denotes the coefficient of έ in p2+1

Let now

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

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


= sin' cosε 0 —


il 3

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

(§n')' + (§'n)' = 2(rr' sin 0 sin 0')' cos 8 (pp)...... (59).

(i-8) (i-8-1)

4 (8 + 1). 1


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13...(8-) (28+1)(28+2)... (28+i-8)


d 1

dz 'd' r

(i − 8) (i − 8 − 1 ) (i − s − 2) (i − s − 3)


4(8 + 1)(8+2).1.2

(that is to say, C=", in accordance with the previous no-
tation,) and let the corresponding notation with accents apply
to '. Then, by the aid of (57), (58), and (59), we have


S1 = 4, cos (84+ a)".


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2i + 1


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Biaxal har monic expressed in symmetrical series of differential


'' coefficients.

" or,





i i

coss(-)... (61), metrical of which, however, the first term (that for which s=0) must be surface halved.

expansion of biaxal


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

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


definite integral investigated.

Cosi '0 sin10 - etc.


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 84. Thus, remem-
bering (16), we have

cos 8


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






(8-) (28+1) (28+2)... (28+i-8)

This holds without exception for the case 80, in which
the second member becomes
It is convenient here to


sin (*) de

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recal equation (44), which, when expressed in terms of "
instead of O


(m, n),


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4, = 24 + 1


where i and 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.

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

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Suppose the co-ordinates 0, & to be used in (52); so that a, 0, $
are the three co-ordinates of P, and we may take do=a2 sin 0d0d.
Working out by aid of (61), (65), the processes indicated
symbolically in (52), we find


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F (0, 4) = Σ {4;~,"' + Σ (4," cos 84 + B sin sø).~“} .


(s) (8)

sin 0 0 0


sin 0() do (i, 8)


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sin Ode

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[** sin s4 F′ (0, 4)do




which is the explicit form most convenient for general use, of the
expansion of an arbitrary function of the co-ordinates 0, & in
spherical surface harmonics. It is most easily proved, [when

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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 dedo, and again by sin so sin do do, and integrating in each case over the whole spherical surface.


(b) In what precedes the expansions of surface harmonics, Review of whether complete or not, have been obtained solely by the differ- expansions entiation of with reference to rectilineal rectangular co- properties.

investigations of

ordinates x, y, z.
The expansions of the complete harmonics
have been found simply as expressions for differential coeffi-
cients, 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

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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.

We find

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


d' V

d'V +

ď2 V +

dx2 dy2 dr

i (i + 1) S; +

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1 d'V sin' do

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Laplace's equation for surface harmonic in polar coordinates.

Definition of "Laplace's functions."

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 0 cos & and sin 0 sin .

(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 0 sin o, it would be the sum of sin (8) terms such as “ integral or fractional, real or imaginary, assume

sp. Now, allowing 8 to have any value



S = (8) 8p



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

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(e) When i and s are both integers we know from §(k) 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 " 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

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