An Elementary Treatise on Spherical Harmonics and Subjects Connected with Them

2

V

2

.

i de

2 de

2

V Hence

de

fa (e) dv,

1=V,V, de

;

fi (e) dV. d.

'

Sf'(e) fi' (e)
V.

V,V,

Uf, (e) fi(e)s

e Now, all over the surface, e=0. Hence

S.f'(o) fi (0)

dS=0. Ifa (0) fi (0)

f'(0) fi'(0) Hence, unless

= 0, which cannot happen

fa (0) fi(0) unless the functions denoted by f, and f, are identical*, or only differ by a numerical factor, we must have

Sjev.v.

'

Now e is proportional to the thickness of the shell at any point. Calling this thickness de, we have therefore

Hence, adding together the results obtained by integrating successively over a continuous series of such surfaces, we get

V,V, dxdydz=0); V, V, now representing solid ellipsoidal harmonics, and the integration extending throughout the whole space comprised within the ellipsoid.

2

* This may be shewn more rigorously by integrating through the space bounded by two confocal ellipsoids, defined by the values 1 and u of €. We then get, as in the text,

folu) film) _fe'() f'}} as=0;

+

Sjevero

Stop

Now the factor within {} cannot vanish for all values of 1 and M, unless the functions devoted by fi and fz be identical, or only differ by a numerical factor,

15. It will be well to transform the expression

fle v,v,as

V,

to its equivalent, in terms of v, u'.

For this purpose we observe that if ds, ds' be elements of the two lines of curvature through any point of the ellipsoid, dS=ds ds'.

Now, ds” is the value of da? + dy' + dz’ when e and v' are constant, ds'?

e and v

(€ + a®) (v + aʻ) (+ aạ) and

;

(6* – a*) (c* — a) therefore if e and u' do not vary,

2dx du

utari

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1

ya .. dso = dx* + dy' + dz' =

4 lia* + 2)2 + 16+)(c* + v).

2

+

Again, differentiating with respect to w the expression

ya za obtained for

-1, we get

c“ + w
ya
22

(u-w) (u' -w)
+
(a + w)? (63+w)(c + w)?
*
(c +w)(a +w) (64 +w) (c* +w)

w
(u w) (e-w) (e-w) (v-w) (u' -w)
+

+
(a’ +w) (6% +w) (c* +w) * (a+ w)(bo+w) (c +w)

+(

++"

2

+

1

E

.. ds?

2

1

S

tu

1

Again, -* *

therefore, putting w=0, ca

(u – v) (e – v) (a + c)2+ (6 + v)*+ (c* + c)2 = (a + v) (b + v) (c + v)

(9 – 1)(e-u)

dv. 4 (a? + v) (+ ) (c* +v) A similar expression holding for ds we get ds? (0-0)* (e-U) (e-)

du du? 16 (a? +u) (6° +v) (c+u) (a + v) (bo+v') (c+)

Y za
,
cm = (a + c)*+(6+e)*+ (6 + e)"

(e- u') ( - )

Ue u)

(a + c) (6* + e) (c +e)' writing e for w in the expression above; ..ed 8? =(a'+e) (62 +€) (c*+e)(u-u)?

dvidu'?. 16 (a +v) (b*+ v) (c*+v) (a + )(b +v') (c*+v) It has been shewn that, integrating all over the surface, the limits of v are - cand , those of v, -68 and - a?.

Hence, V, V,, denoting two different ellipsoidal harmonics

V,V, (u'-u) dvdu
--> {(ao+v) (bo+u) (C+0) (ao+v) (6P+0] (C+v')}}
The value of the expression slfv-dxdyda

, or its equivalent

V2 (u'-u) dvdu abc

-«2{(a+u) (6° +v) (c+v) (a +v) (6+') (c*+v')}}' in any particular case, is most conveniently obtained by expressing V as a function of x, y, z.

1

0.

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16. Before proceeding further with the discussion of ellipsoidal harmonics in general, we will consider the special case in which the ellipsoid is one of revolution. We must enquire what modification this will introduce in the quantities which we have denoted by a, b, y, viz.

a=

= 0.

dß?

(a” + )(6+4)*(0* + 4)

dy
(a” + ) (+4)* (co + %)*'

dy

? (a) and in the differential equation

ď? V
đV

TV
(u – u) dax +(v'—e)

u da2

+ (e – u)

dy2 We will first suppose the axis of revolution to be the greatest axis of the ellipsoid, which is equivalent to supposing ' =c%. To transform a and y, put a2 + y = 0%, a' terno, a’ +v' = w'; we then obtain

de

1 n+ – a b)

log

n-a– ' de

1 (a– 62). – W

log 02 – a’ + b2 (a’ – 6?)?

(a’ – °)4 + w To transform ß, we must proceed as follows.

Put y=-o cos' - Usina, v=-c cos' °-b sind, we then get generally b? + y = (b? — c) cos'a, c* +4=(c* — 6%) sin'w;