The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 14 of 39

whilst by the addition of successive equations of the type of (16-1), we obtain

3-Pn = (2n-l)Pn-l + (2n-5)Pn^+ (166).

0(1

222 Methods for the Solution of Special Problems [ch. viii

254. We  have  had  the  general  theorem  (§  237) 
     

jjsnsmda> = o,

from which the theorem

JJpn(ji)Pm(M)dOi = 0

follows as a special case. Or since

dco = sin 6d0d<j> = — d/idfy,

j+1 Pn^)Pm{ji)dli = Q (167).

To find I Ftf (/Jb) dfi, let us square the equation

o multiply by d/x, and integrate from /* = — 1 to yu. = + 1. The result is

r+i <*> J -l 0

" + 1 <»

-l 0

all products of the form PnPm vanishing on integration, by equation (167).

Thus I Pndfi is the coefficient of A2K in

[+1 dp

J -i 1 -

i.e. in

2/i/x + A» ' 1, 1 - /i

9

and this coefficient is easily seen to be — We accordingly have

£{««}•*- £^1 d<58).

255. We  can  obtain  this  theorem  in  another  way,  and  in  a  more  genera]  form,  by 
     

using the expansion of § 240, namely

FpBSdd> ! (2s + 1} jjFF* (cos ^ dS>

where 6 is the angle between the point P and the element dS on the sphere. This expansion is true for any function F subject to certain restrictions. Taking F to be a surface harmonic Sn of order n, we obtain

(^„)p = ^-2T(25 + 1) ( [snPt (cos 6) dS

«=o In 4

^f J SnPn (cos 6) dS,

254-25G] Spherical Harmonics 223

all other integrals vanishing by the theorem of § 237. Thus

//

AM^i^OU*-!

or

JfsnPn(fjL)da> = ~-(Sn)^l (169).

This is the general theorem, of which equation (168) expresses a particular case. To pass to this particular case, we replace Sn by Fn (/x) and obtain, instead of equation (169),

ff{Pn (M)F sin dd6d4>= ~ Pn (1),

or, after integrating with respect to <f>,

agreeing with equation (168).

Expansions in Legendre's Coefficients.

256. Theorem. The value of any function of 6, which is finite and single-valued from 6 = 0 to 6 = ir, and which has only a finite number of discontinuities and of maxima and minima within this range, can be expressed, for every value of 0 within this range for which the function is continuous, as a series of Legendre's Coefficients.

This is simply a particular case of the theorem of § 240. It is therefore unnecessary to give a separate proof of the theorem.

The expansion is easily found. Assume it to be

/(//,) = a0 + a1Pl + a2P2 + ... + asPs + (170),

then on multiplying by P„,{ii)dfi, and integrating from /* = — 1 to /x, = + l, we obtain

r+i s=oo r+i

I

pn (a0/(/) dfil= Z a8 Ps(p)Pn (/) dfj,

s=0 J —l

— 1 S = 0 J —1

la,

2n + 1 ' every integral vanishing, except that for which s = n. Thus

2w + 1 r+i

a*

jy.i^fi^dfM (i7i),

2

giving the coefficients in the expansion.

If f(fjb) has a discontinuity when /jl = /j,0, the value assumed by the series (168) on putting /j, = /*<, is, as in § 240, equal to

i WW +/■ «} (172),

where fi(/J>0), /3(/x0) are the values of f(/j.) on the two sides of the discon- tinuity.

224 Methods for the Solution of Special Problems [oh. viii

Harmonic Potentials.

257. We  are  now  in  a  position  to  apply  the  results  obtained  to  problems 
     

of electrostatics.

Consider first a sphere having a surface density of electricity Sn. The potential at any internal point P is

'Snds rr snds

VP =

f[SndS_[[ J J PQ JJV5

-//

2 — 2ar cos 6 + r2 ^ ( 1 + - £(cos 6) + -* £(cos 0)+...)dS

b7T

In + 1 an+1 4tt rnSn

0,2 ~^+i(Sn)coso=i> by the theorems of §§237 and 255,

.(173),

2n + 1 a™

this expression being evaluated at P.

Similarly the potential at any external point P is

4,7ran+2Sn p (2n + l)r»+1'

These potentials are obviously solutions of Laplace's equation, and it is easy to verify that they correspond to the given surface density, for

\ 3^ /outside <W

inside

This gives us the fundamental property of harmonics, on which their application to potential-problems depends • A distribution of surface density Sn on a sphere gives rise to a potential which at every point is proportional to Sn.

258. The density of the most general surface distribution can, by the theorem of § 240, be expressed as a sum of surface harmonics, say

a — S0 + Si + S.j + ...,

in which S0 is of course simply a constant. The potential, by the results of the last section, is

( S /r\ S /r\2 ) V = 4<'7ra\S0 + ~{-j +-^i-j +...Y at an internal point ...(174),

= 4?rtt \S0 (-) + oM-) + -f(-) +•••[ at an external point ...(175).

257-259]

Spherical Harmonics

225

Examples of the use of Harmonic Potentials.

I. Potential of spherical cap and circular ring.

259. As  a  first  example,  let  us  find  the  potential  of  a  spherical  cap 
     

of angle a — i.e. the surface cut from a sphere by a right circular cone of semivertical angle a — electrified to a uniform surface density <x0.

We can regard this as a complete sphere electrified to surface density a, where

a- = a0 from 6 = 0 to 6 = a, a=0 from 6 = a to 6 = ir.

The value of a being symmetrical about the axis 6 = 0, let us assume for the value of a expanded in harmonics

a = a0 + axI^ (cos 6) + aiJ^ (cos 6) + ... then, by equation (171),

2n + l f9=°

Fig. 76.

an =

2

2w + l

aPn (cos 6) d (cos 6)

[0 = 0

Pn (cos 6) d (cos 6)

J 9 = a

= J <r0 [Pn-i (cos a) - Pn+1 (cos a)} by equation (165), except when n =0. For this case we have

0 = 0

«o = \ °"o f d (cos 6) = J o-0 (1 — cos a).

0 = a

Thus

h<To

(1 - cos a) + 2 J Pn-i (cos a) - i^+1 (cos a) [• 2J (cos 6) j

»=H J J

It is of interest to notice that when 6 = a, the value of a given by this series is a=^cr0) as it ought to be (cf. expression (172)).

The potential at an external point may now be written down in the

a - cos .) (?) + T u»^(»^) £p. (cos 9)-

(176),"

form

V= 2iraa0

and that at an internal point is V = 2iraa0

(i - cos «) + T JL-E" «>-*«(«■«> fry ■

tt = l Ztt + 1 W

.(177). 15

226 Methods for the Solution of Special Problems [oh. viii

On differentiating with respect to a, we obtain the potential of a ring of line density <r0adcc. At a point at which r > a, we differentiate expression (176), and obtain

-1+ 2 i^ (cos a) sin a (-] i£(cos#) I,

or, putting acr0da = t and simplifying,

F=2ttt 2 Pn (cos a) sin a (-) Pn(cos6) .(178).

n = 0 v*/

Obviously the potential at a point at which r < a can be obtained on

replacing (^ by Q .

260. These  last  results  can  be  obtained  more  directly  by  considering 
     

that at any point on the axis 0 = 0 the potential is

2iraT sin a

or, if r > a,

V = -, „

vr2 H- a2 — 2ar cos a

27rar sin a n=°°

YP» (cos «)(£)*,

n=0 v /

and expression (178) is the only expansion in Lagrange's coefficients which satisfies Laplace's equation and agrees with this expression when 6 = 0.

II. Uninsulated sphere in field of force.

261. The method of harmonics enables us to find the field of force produced when a conducting sphere is introduced into any permanent field of force. Let us suppose first that the sphere is uninsulated.

Fig. 77.

259-261] Spherical Harmonics 227

Let the sphere be of radius a. Round the centre of the field describe a slightly larger sphere of radius a, so small as not to enclose any of the fixed charges by which the permanent field of force is produced. Between these two spheres the potential of the field will be capable of expression in a series of rational integral harmonics, say

V=V0 + V1+Vi + (179).

The problem is to superpose on this a potential, produced by the

induced electrification on the sphere, which shall give a total potential

equal to zero over the sphere r = a. Clearly the only form possible for this new potential is

r—®-®'-*@'- <180>-

Thus the total potential between the spheres r = a and r = a' is

Putting Vn = rnSn, the surface density of electrification on the sphere is, by Coulomb's Law,

S(2ti + 1)K.

4<7ra

This result is indeed obvious from § 258, on considering that the surface electrification must give rise to the potential (180).

If n is different from zero,

fjsoSndS = Q, where the integration is over aDy sphere, so that

JJsndS = 0 (n^O),

and ffVndS = 0 (n^O) (181).

Thus the total charge on the sphere

= --j V0 . kira? = - V0a,

4>7ra

and T^ was the potential of the original field at the centre of the sphere.

15—2

228 Methods J "or the Solution of Special Problems [ch. viii

262. Incidentally we may notice, as a consequence of (181), that the mean value of a potential averaged over the surface of any sphere which does not include any electric charge is equal to the potential at the centre (cf. § 50).

If the sphere is introduced insulated, we superpose on to the field

already given, the field of a charge E spread uniformly over the surface of

E the sphere, and the potential of this field is — . We obtain the particular

case of an uncharged sphere by taking E = VQa, and the potential of this

field, namely Io(-)> just annihilates the first term in expression (180), to

which it has to be added.

It will easily be verified that, on taking the potential of the original field to be Vi=Fx, we arrive at the results already obtained in § 217.

III. Dielectric sphere in a field of force.

263. An analogous treatment will give the solution when a homo- geneous dielectric sphere is placed in a permanent field of force. The treatment will, perhaps, be sufficiently exemplified by considering the case of the simple field of potential

V1 = Fx = rS1. Let us assume for the potential VQ outside the sphere

Fig. 78.

and for the potential Vi inside the sphere

Vi = l3rS1} do term of the form -J being included in Vi, as it would give infinite

262-264] Spherical Harmonics 229

potential at the origin. The constants a, /3 are to be determined from the conditions

Vi = X I ira^==a^Ut r = a.

dr dr J

These give a + -=/3a,

(Jj

a3 whence a = - -^ — ~ a3, /3 =

so that V0 = Fx\l-j

K + 2™' H~K + 2'

K-l [c£3^ r,

V< = KT2F*-

Thus the lines of force inside the dielectric are all parallel to those of the original field, but the intensity is diminished in the ratio ^ — = . The field is shewn in fig, 78.

IV. Nearly spherical surfaces.

264. If r = a, the surface r = a + x> where % is a function of 6 and <f>, will represent a surface which is nearly spherical if ^ is small. In this case % may be regarded as a function of position on the surface of the sphere r = a, and expanded in a series of rational integral harmonics in the form

X = S0 + S1 + S.i + ...

in which Slt S2, ... are all small.

The volume enclosed by this surface is

£ j j r3do)

= i (a3 + Sa*x) dec

47ra3

00

4}ira3 . „ ~ = — s- + 4-Tra2 S0.

If $o = 0, the volume is that of the original sphere r = a.

230 Methods for the Solution of Special Problems [en. vin

The following special cases are of importance :

r = a + eP^ To obtain the form of this surface, we pass a distance e cos 6 along the radius at each point of the sphere r = a. It is easily seen that when e is small the locus of the points so obtained is a sphere of radius a, of which the centre is at a distance e from the origin.

rsa + ajSj. The most general form for a^ is Ix + rny + nz, and this may be expressed as ae cos 6, where 6 is now measured from the line of which the direction cosines are in the ratio I : m : n. Thus the surface is the same as before.

r = a + S2. Since r is nearly equal to a, this may be written

r2=a2+2aS2

2

= a2+-r2S2, a

or a? + y2 + z2 = a2 + an expression of the second degree.

Thus the surface is an ellipsoid of which the centre is at the origin. It will

easily be found that r = a + eP2 represents a spheroid of semi-axes a -j- e, a — ^ ,

3e

and therefore of ellipticity ~- .

265. We can treat these nearly spherical surfaces in the same way in which spherical surfaces have been treated, neglecting the squares of the small harmonics as they occur.

266. As an example, suppose the surface r = a + Sn to be a conductor, raised to unit potential. We assume an external potential

A ™ fa\n+1

r \rj

where A and B have to be found from the condition that V=l when r = a+8n- Neglecting squares of Sn, this gives

A / Sn

a\ a)

so that A = a, B = ~,

ct

a an

and V = - + —-T-. Sn.

/v% ,^71+1

By applying Gauss' Theorem to a sphere of radius greater than a we readily find that the total charge is a, the coefficient of -. Thus the

264-267] Spherical Harmonics 231

capacity of the conductor is different from that of the sphere only by terms in Sn2, but the surface distribution is different, for

a dV W •<• 1 u o.

4)7ro- = — ^— = — -z— , it we neglect £L2>

on or °

• £+(«+*) ;Ib «•

a2\ a / V a2

1 TO-1

a a"5

the surface density becoming uniform, as it ought, when n = l, i.e. when the conductor is still spherical.

267. As a second example, let us examine the field inside a spherical condenser when the two spheres are not quite concentric. Taking the centre of the inner as origin, let the equations of the two spheres be

r = a,

We have to find a potential which shall have, say, unit value over r = a, and shall vanish over r = b + ei?. Assume

V^ + i^+C + DZr, r r2

when B and D are small, then we must have

These equations must be true all over the spheres, so that the coefficients of i? and the terms which do not involve T\ must vanish separately. Thus

• • C-1 = 0; -+Da = 0;

a a2

j+g=°> -e4+¥+Db=o-

From the first two equations

ab A=- ,

b — a

and this being the coefficient of - in the potential, is the capacity of the

condenser. Thus to a first approximation, the capacity of the condenser remains unaltered, but since B and D do not vanish, the surface distribution is altered.

232 Methods for the Solution of Special Problems [ch. vni

.

Y. Collection of Electric Charges.

267 a. If a collection of electric charges are arranged in any "way whatever subject only to the condition that none of them lie outside the sphere r = a, then the potential at any point outside the sphere must be

-p. e Si S2

' — "1 « "I ; T ••• J

where e is the total charge inside the sphere (cf. § 266) and Slt &>, ... are surface harmonics which depend on the arrangement of the charges inside the sphere.

If the total charge is not zero, the potential can also be treated as in § 67, and on comparing the two expressions obtained for the potential, we can identify the harmonics S1}S2,.... We find that

and it will be easily verified by differentiation that the expressions on the right are harmonics.

This example is of some interest in connection with the electron-theory of matter, for a collection of positive and negative charges all collected within a distance a of a centre may give some representation of the structure of a molecule. The total charge on a molecule is zero, so that we must take e = 0, and the potential becomes

The most general form for St is (cf. § 239) -(Ax+By + Cz), or n cos 6, where 6 is the

angle between the lines from the origin to the point x, y, z and that to the point At B, 0 &ndixisJ(A2+B2 + C2).

. , . u cos 6 , Thus the term which is important in the potential when r is large is - — ^ — ? shewing

that at a sufficient distance the molecule has the same field of force as a certain doublet of strength /*. Clearly when fi has any value different from zero, the molecule is "polarised" (cf. § 142) in Faraday's sense. If /x = 0,the potential becomes

shewing that the force now falls off as the inverse fourth power of the distance.

It is worth noticing that the average force at any distance r is always zero, so that to obtain forces which are, on the average, repulsive, we have to assume the presence of terms in the potential which do not satisfy Laplace's equation, and which accordingly are not derivable from forces obeying the simple law e/r2 (cf. § 192).

267a-269] Spherical Harmonics 233

Further Analytical Theory of Harmonics. General Theory of Zonal Harmonics.

268. The general equation satisfied by a surface harmonic of order n, which is symmetrical about an axis, has already been seen to be

k{(1-^}+n(n+1)Sn=0 (182)'

One solution is known to be Pn, so that we can find the other by a known method. Assume Sn = Pnu as a solution, where u is a function of ix. The equation becomes

(l-",)|i{i"+-p»|}-2'i{|"+p»|}+»(»+l)'p»M=0-(183)'

and, since Pn is itself a solution,

(W')|;(g)-V§ + »(» + l)ii=0.

Multiplying this by u and subtracting from (183), we are left with

or, multiplying by Pn and rearranging,

or again £{<!-*> *) g + {d - rf> SI 1 g) - 0.

On integration this becomes

(1 - u?) Pn* ^ = constant.

OfM

We may therefore take in which the limits may be any we please. If we write

«»=<c™ (184)-

the complete solution of equation (182) is

Sn=Pnu = APn + BQn.

269. The two solutions Pn and Qn can be obtained directly by solving the original equation (182) in a series of powers of /a.

Assume a solution

Sn = bofS + blfx^ + b^+* + . . . ,

234 Methods for the Solution of Special Problems [ch. viii

substitute in equation (182), and equate to zero the coefficients of the different powers of fi. The first coefficient is found to be b0r(r— 1), so that if this is to vanish we must have r = 0 or r = 1. The value r = 0 leads to the solution

n(w + l) , , (n-2)w(n+l)(w+3)..4 Uo-i-- 12 fi + 1.2.3.4 ^ '"

while the value r = 1 leads to the solution

(n - 1) (n + 2) (n - 3) (n - 1) (n + 2) (n + 4)

Wl_/t EO ^ + 1.2.3.4.5 ** -•

The complete solution of the equation is therefore

au0+ /3WJ.

If n is integral one of the two series terminates, while the other does not. If n is even the series u0 terminates, while if n is odd the terminating series is w,. But we have already found one terminating series which is a solution of the original equation, namely Pn. Hence in either case the terminating series must be proportional to 1^, and therefore the infinite series must be proportional to Qn.

270. We can obtain a more useful form for Qn from expression (184). The roots of i^ (fi) = 0 are, as we have seen, n in number, all real and separate, and lying between — 1 and + 1. Let us take these roots to be a1} a2, ... an. Then

1 u + 1 via - a. (u,- a.)V v

fi - 1 fi + 1 \fi-a, (jj, - asyj

on resolving into partial fractions. Putting fj, — + 1 and — 1, we find at once that a = , & = — £.

In the general fraction

1 1

D {x — al){x — ai)...i

let us suppose all the factors in the denominator to be distinct, so that we may write

C-% Co

• — — + ....

D x— Ox x — a2 On putting # = &!, we obtain at once

0i =

{ax - a,) (a, - a,) (a, - a4) . .. ' 1

C2 = ; r- — - , etC

(a2 — a2) (a, - a3) (a2 — a4) . . .

Spherical Harmonics

235

Now let aa and <za become very nearly equal, say aa = ax + dux> then

1

c,= -

da-i (a1 — a3)(a1— a4)...'

while

The fractions now combine into

Co =

da,! (a2 - a3) (<x2 — a4) ... '

Cl , C2

(ci + c2) # - (ci a2 - c^) and on putting this equal to

Cn Co

x — ax (x — a-,)- ' it is clear that the value of c/ must be taken to be cx + c2. Now

2 da-L ((a2-a3)( 1 (3

-iif

1 I 1

a2 — a4) . . . (oj — a3) («i — at) ...)

da-, [dx (# - a,) (a? - a4) .../»=«,

3*1 £ JW

and this remains true however many of the roots a3, a4 ..., coincide among themselves, so long as they do not coincide with the root Oj. Thus, in expression (185), the value of cg is

3 f Q*-«.)» ]

Putting we find that

•%(/) /-«•

= £(/*)»

c8 =

.if.

a/. l(i - ^) {is wpj „,„, s«, Ki - «.•) (B («.))•; •

Since (/a — ag) i£ (/*) is a solution of equation (182), we find that

^ [(1 - ^R 00 + 0* - «.) ^}] + n (n + 1) 0* - «.) -B = 0. On putting fi — a8, this reduces to

da.

{(1

aa>)B(a8)}+(l-«s>)d4^ = 0,

das

giving, on multiplication by R (as),

~[d-^){R(as)Y] = 0.

Hence cs = 0.

236 Methods for the Solution of Special Problems [ch. vui

Equation (185) now becomes 1

0*-i){&00}--]

so that, on integration,

fl— 1 fM+1

• x

d„

F

= £log^ + £

ds

.(186),

(/» - 1) {Pn (ft)} »"-°/-l- /-<% On multiplying by Pn{fj)} we obtain from equation (184),

where Wn-! is a rational integral function of /j, of degree n — 1.

It is now clear that Qn (fi) is finite and continuous from /*=— 1 to /a = + 1, but becomes infinite at the actual values fi = + 1.

To find the value of T^_i we substitute expression (186) in Legendre's equation, of which it is known to be a solution, and obtain

9 {(1-^)^1 + ^+1)1^

3/4

--$&-»&

^^og^)yn(n+l)^Pn(^log^±\

= 2

dp,

= 2{(2n-l)Pn-1 + (2n-5)Pn-3+...} (187).

Since Wn-i is a rational integral algebraic function of /x of degree n — 1, it can be expanded in the form

"n— l = &\Pn—\ H" a2-*n— 2 + ••• 5

so that

9 1(1-^)^1 + n(n + 1)1^

= %ag

1 {(1 - ^) %sj + n (n + 1) i?,_s

_9yU, ( 3/1

= 2a« (n (n + 1) — (w - s) (n - 5 + 1)} i^_g.

Comparing with (187), we find that as = 0 when s is odd, and is equal to

2(2?i-2s+l) s(2n-s + l) when s is even.

Thus

w 2n-lp 2tt-5 p 2n-9

and

Qn = £-& (/*) log

'"_I ' 3(n-l)

/t+1 2ft -1

/*

1.1*

-*w— l

5 (» - 2)

2w-5 3(w-l)

-*Tl— 3 t • • •

270-273] Spherical Harmonics 237

271. When we are dealing with complete spheres it is impossible for the solution Qn to occur. If the space is limited in such a way that the infinities of the Qn harmonic are excluded, it may be necessary to take into account both the l?n and Qn harmonics. An instance of such a case occurs in considering the potential at points outside a conductor of which the shape is that of a complete cone.

Tesseral Harmonics.

272. The equation satisfied by the general surface harmonic Sn is

sin 090 V oU J sm2 6 d<f>-

As a solution, let us examine

sn = ©s>,

where © is a function of 6 only, and <l> is a function of $ only. On

substituting this value in the equation, and dividing by @<£/sin2 6, we obtain

sin 6 d ( . ad®\ 1 32<E> , , , - x . , a A

We must therefore have

Id2® _

<£> 8</>2 " *' sin 6 d ( . . 9©\ , . -, x • 2 /,

The solution of the former equation is single valued only when k is of the form — m2, where m is an integer. In this case

<E> = Cm cos mc/> + Dm sin m<£, and © is given by

1 d ( . a a©\ f , , x m2 ) _ _ sin-^(Sm^8^) + r(?l + 1)-sin^|0 = O>

or, in terms of fi,

4^-^S+i"(-+1)-Ale-° (188)'

an equation which reduces to Legendre's equation when m = 0.

273. To obtain the general solution of equation (188), consider the differential equation

(1-^)^+2/^ = 0 (189),

of which the solution is readily seen to be

z=C(-ti?)n (190).

If we differentiate equation (189) s times we obtain

238 Methods for the Solution of Special Problems [ch. vin

If in this we put s = n, and again differentiate with respect to p, we obtain

hfr-K®}**1®- (192)'

dnz which is Legendre's equation with ^— - as variable. Thus a solution of this

equation is seen to be

giving at once the form for Pn already obtained in § 249. The general solution of equation (192) we know to be

d^n = APn + BQn.

If we now differentiate (192) m times, the result is the same as that of differentiating (189) m + n+1 times, and is therefore obtained by putting s = m + n+ 1 in (191). This gives

(l-/^|-^-2(m + l)^^^

m

or, multiplying by (1 — /a2) 2 , 0- - V?)* o,.m+n+2 - 2 (m + 1) /* (1 - /x2)2-

!» 2m+n »

• (m + ri + l)(7i-m)(l-^2)2^T-n = 0 (193).

Let (l-/x2)2|— - - = ».

Then * (l-^.£-* ^(i-^.

*r "'^H1-^ p^-(2™+2)M-^ 5;

1^

a^ £{a-io£}-(i-^JS-^+^a-^K

2; »_■/) 3m+n-.

■ ffl (l-/i')s-^(l -JU2)2

= - i; j(m + n + 1) (n - m) + m - ,^7— ,f > by equation (193),

= — v Ui(n+ 1) —

m+n

2, ,2

?n2

1 -/rj Thus v satisfies

and this is the same as equation (188), which is satisfied by ®.

273, 274] Spherical Harmonics 239

274. The  solution  of  equation  (188)  has  now  been  seen  to  be 
     

© = (1 - a2) 2 - -

where . ^—n = APn + BQn.

m

dmP -?mO

Hence . 0 = ^(1-^)^ + 5(1-^^

The functions ( W)2 ^, (W»)2^?

are known as the associated Legendrian functions of the first and second kinds, and are generally denoted by P% (/x), Q™ (ft). As regards the former we may replace Pn, from equation (159), by

1 dn — — — (u? - 1V»

and obtain the function in the form

1 VI 7m+n

p"(^)=2^(i-^6^2-i)n (194)-

It is clear from this form that the function vanishes if m + n > 2n, i.e. if m > n. It is also clear that it is a rational integral function of sin 6 and cos 6. From the form of Qn (/j.), which is not a rational integral function of ll, it is clear that Q™ (/x) cannot be a rational integral function of sin 6 and cos 6.

Thus of the solution we have obtained for Sn, only the part

P£ (fi) (Cmcos m<f> + Dm sin m$) %

gives rise to rational integral harmonics. The terms P™ (/x) cos m<f) and P™ (/a) sin m(f> are known as tesseral harmonics.

Clearly there are (2n + 1) tesseral harmonics of degree n, namely

Pn(fi), cos <f>Pl(fi), sin 4> P(ji), ... cosw^PJO*), sin n#P£(/4

These may be regarded as the (2n + 1) independent rational integral har- monics of degree n of which the existence has already been proved in § 239.

Using the formula

and substituting the value obtained in § 247 for Pn(fi) (cf. equation (155)), we obtain P™ (/x) in the form

(2n) ! sin- fl f n_m _ (n-m)(n-m-l) n_m_2 ^W = 2»W!(n-m)ilC°S * 2(2»-l)

(» - to) (n - to- 1) (n -m-2) (n-m-3) B_jn_4 . _ 1

• 2.4(2»-l)(2w-3) "T

240 Methods for the Solution of Special Problems [ch. viii

The values of the tesseral harmonics of the first four orders are given in the following table.

Order 1. cos 0, sin 0 cos 0, sin 0 sin 0.

Order 2. £(3cos30 — 1), 3 sin 0 cos 0 cos 0, 3 sin 0 cos 0 sin 0,

3 sin2 0 cos 20, 3 sin2 0 sin 20.

Order 3. £ (5 cos3 0 - 3 cos 0), f sin 0 (5 cos2 0 - 1) cos 0,

•I sin 0 (5 cos2 0 — 1) sin 0, 15 sin2 0 cos 0 cos 20, 15 sin2 0 cos 0 sin 20, 15 sin3 0 cos 30, 15 sin3 0 sin 30.

Order 4. £ (35 cos" 0-30 cos2 0 + 3), § sin 0 (7 cos3 0 - 3 cos 0) cos 0, | sin 0 (7 cos3 0 - 3 cos 0) sin 0, jy* sin2 0 (7 cos2 0-1) cos 20, -V5- sin2 0 (7 cos2 0 - 1) sin 20, 105 sin3 0 cos 0 cos 30, 105 sin3 0 cos 0 sin 30, 105 sin4 0 cos 40, 105 sin4 0 sin 40.

275. We  have  now  found  that  the  most  general  rational  integral  surface 
     

harmonic is of the form

n

Sn = %P% (ft) (Am cos m0 + Bm sin m0), o

in which P™(/a) is to be interpreted to mean i^(/i), when m = 0.

Let us denote any tesseral harmonics of the type

P™(/i)(.4cosm0 + Psinm0) by S™.

Then by § 237, jl S% fl™, day = 0

if n ={= «'. If w = w', then

JJ8S fl* = // T (A) *?' 00 (4» cos m0 + flm sin m0)

(.4m' cos m 0 + i?^' sin m' 0) d&>, and this vanishes except when m = m'.

When n = n' and m = ra' the value of 1 1 S% S%f dw clearly depends on that of I [P™ (fi)}2 dfi, and this we now proceed to obtain.

We have

r+l r+l /pirn p \ 2

J.i1^0*)N/'=j_i(i-/*,)B(y **

(1 - ^)'

274-276] Spherical Harmonics 241

dnz Since ^— - = i?t is a solution of equation (191), we obtain, on taking s = to + n

in this equation, and multiplying throughout by (1 — fM2)m~\

dm~1P

• (n + to) (w - to + 1) (1 - ya2)"1"1 g-^?,

which, again, may be written

In equation (195) the first term on the right-hand vanishes, so that

f+l r+l /^m-ip\2

J _t {P- (/)} dp = (n + to) (n - to + 1) J ^ (1 - ^r-1 (-g-^J rf/*

' = (n + to) (n - to + 1) J** {P™-1 0")}2 ^, a reduction formula from which we readily obtain

" (p» «)■ «, = ((^™| ;/*' (p„ «}> *.

2 (n + m) !

2/« + 1 (n — to) !" These results enable us to find any integral of the type J 1 $nS'n c?a>.

Biaxal Harmonics.

276. It is often convenient to be able to express zonal harmonics referred to one axis in terms of harmonics referred to other axes — i.e. to be able to change the axes of reference of zonal harmonics.

Let ^ be a harmonic having OP as axis. At Q the value of this is Pn (cos 7), where 7 is the angle PQ, and our problem is to express this harmonic of order n as a sum of zonal and tesseral harmonics referred to other axes. With reference to these axes, let the coordinates of Q be 6, ty, let those of P be ©, <£, and let us assume a series of the type

s = n

Pn (cos 7) = 2 P"n (cos 6) (As cos sty + Bs sin sty).

Let us multiply by Psn (cos 6) cos sty and integrate over the surface of a unit sphere. We obtain

J [pn (cos 7) [Psn (cos 6) cos sty) do = A, J J {Pn (cos 0)}2 cos2 sty dco. j. 16

242 Methods for the Solution of Special Problems [ch. viii

By equation (169),

J J Pn (cos 7) {Psn (cos 6) cos s<J>] dco = ^ ^ {P* (cos 0) cos s<£jy=0

= 2^1 P» (cos ®) cos s®, and J|{P; (cos 0)}2 cos2 s<f> do = j+l{P°n (/x)}2 dp [^ cos2 50 d<j>

Thus

2?r (n + g) ! 2n + 1 (n - s) ! '

(r> — <A I -4. = 2 i_ ^; pj (cos 0) cos 5$, (w + s) !

and similarly

^ = 2(^|-;P«(cos@)SmS«>.

This analysis needs modification when s = 0, but it is readily found that

4o = £(cos0), Pn = 0,

so that

P» (cos 7) = Pn (cos 0) P„ (cos 0) +T 2 ^—4-; P* (cos 0) P^ (cos 0) cos s (6 - <£)

«=i (?i + s) !

(196).

General Theory of Curvilinear Coordinates.

277. Let  us  write 
     

0 0> 2/, *) = \ yfr (x, y, z) = p,

X 0> V> z) = v> where $, yjr, ^ denote any functions of x, y, z. Then we may suppose a point in space specified by the values of X, /j,, v at the point, i.e. by a knowledge of those members of the three families of surfaces

</> (x, y, z) — cons. ; ^ (x, y, z) = cons. ; % (x, y, z) = cons.

which pass through it.

The values of X, ft, v are called " curvilinear coordinates " of the point. A great simplification is introduced into the analysis connected with curvilinear coordinates, if the three families of surfaces are chosen in such a way that they cut orthogonally at every point. In what follows we shall suppose this to be the case — the coordinates will be " orthogonal curvilinear coordinates."

The points X, /x, v and X + dX, p, v will be adjacent points, and the distance between them will be equal to dX multiplied by a function of

276-278] General Curvilinear Coordinates 243

X, /a, and v — let us assume it equal to y- • Similarly, let the distance

ft]

from A,, fx, v to X, /x + d/x, v be -7- , and let the distance from X, /x, v to

rfra

7 i_ ' ^"

A,, u, *> + af be -j— .

Then the distance ds from X, /x, v to X + d, /x + d/x, v + dv will be given by

this being the diagonal of a rectangular parallelepiped of edges

dX d/x 1 dv fti ' ft2 ^3

Laplace's equation in curvilinear coordinates is obtained most readily by applying Gauss' Theorem to the small rectangular parallelepiped of which the edges are the eight points

X ± \ dX, 1^+2 dfi> v + \ dv.

In this way we obtain the relation

•dV

in the form

//

n<® = 0 (197)

dX\h.2h3 oX J dix\h3hidfx) dvX^h* dv J

and as we have already seen that equation (197) is exactly equivalent to Laplace's equation V2V = 0, it appears that equation (198) must represent Laplace's equation transformed into curvilinear coordinates.

In any particular system of curvilinear coordinates the method of pro- cedure is to express h^, h2, h3 in terms of X, /x and v, and then try to obtain solutions of equation (198), giving V as a function of X, fx and v.

Spherical Polar Coordinates.

278. The system of surfaces r = cons., 6 = cons., cf> = cons, in spherical polar coordinates gives a system of orthogonal curvilinear coordinates. In these coordinates equation (198) assumes the form

dr V dr)+ sin 6 BO \Sm dO J + sins 0 dp ~ '

already obtained in § 233, which has been found to lead to the theory of spherical harmonics.

1G— 2

244 Methods for the Solution of Special Problems [oh. vm

Confocal Coordinates.

279. After spherical polar coordinates, the system of curvilinear coordi- nates which comes next in order of simplicity and importance is that in which the surfaces are confocal ellipsoids and hyperboloids of one and two sheets. This system will now be examined.

Taking the ellipsoid

as a standard, the conicoid

x2

l+l2-^1 "d")

1-1^+^ = 1 (200)

a- + 0 b2 + 6 c2 + 0

will be confocal with the standard ellipsoid whatever value 6 may have, and all confocal conicoids are represented in turn by this equation as 8 passes from — oo to + co .

If the values of x, y, z are given, equation (200) is a cubic equation in 6. It can be shewn that the three roots in 6 are all real, so that three confocals pass through any point in space, and it can further be shewn that at every point these three confocals are orthogonal. It can also be shewn that of these confocals one is an ellipsoid, one a hyperboloid of one sheet, and one a hyperboloid of two sheets.

Let A, fi, v be the three values of 6 which satisfy equation (200) at any point, and let A, fi, v refer respectively to the ellipsoid, hyperboloid of one sheet, and hyperboloid of two sheets. Then A, /u,, v may be taken to be orthogonal curvilinear coordinates, the families of surfaces A = cons., lc = cons., v = cons, being respectively the system of ellipsoids, hyperboloids of one sheet, and hyperboloids of two sheets, which are confocal with the standard ellipsoid (199).

280. The first problem, as already explained, is to find the quantities which have been denoted in § 277 by h1} 7i2, hs. As a step towards this, we begin by expressing x, y, z as functions of the curvilinear coordinates A, lc, v.

The expression

^2 /i»2 /y2