A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 13 of 28

satisfies Laplace's equation, therefore every function formed from this by differentiation with respect to any number of axes in suc- cession must also satisfy that equation.

A point of the first order may be formed by taking two points of order zero, having equal and opposite charges — AQ and AQ) and placing the first at the origin and the second at the extremity of the axis h^. The value of h^ is then diminished and that of A0 increased indefinitely, but so that the product A0 h^ is always equal to Alt The ultimate result of this process, when the two points coincide, is a point of the first order whose moment is A1 and whose axis is 7^. A point of the first order is therefore a double point. Its potential is

By placing a point of the first order at the origin, whose moment is — A19 and another at the extremity of the axis h% whose moment is Alt and then diminishing ^2 and increasing A19 so that

182 SPHERICAL HARMONICS. [129 d.

we obtain a point of the second order, whose potential is

£--**,-- 7

<

We may call a point of the second order a quadruple point because it is constructed by making four points of order zero ap- proach each other. It has two axes h-± and k% and a moment A2. The directions of these axes and the magnitude of the moment completely define the nature of the point.

By differentiating with respect to n axes in succession we obtain the potential due to a point of the nih order. It will be the product of three factors, a constant, a certain combination of cosines, and r (n+1). It is convenient, for reasons which will appear as we go on, to make the numerical value of the constant such that when all the axes coincide with the vector, the coefficient of the moment is r(n+1\ We therefore divide by n when we differ- entiate with respect to hn.

In this way we obtain a definite numerical value for a particular potential, to which we restrict the name of The Solid Harmonic of degree — (?£+ 1), namely

1 d d

1.2.3. ..nd^ dhz dhn r

If this quantity is multiplied by a constant it is still the poten- tial due to a certain point of the nih order.

129 d.] The result of the operation (13) is of the form

F= Ynr~(n+l\ ' (14)

where Tn is a function of the n cosines fa... i*n of the angles between r and the n axes, and of the \ n (n—l) cosines A12, &c. of the angles between pairs of the axes.

If we consider the directions of r and the n axes as determined by points on a spherical surface, we may regard Yn as a quantity varying from point to point on that surface, being a function of the \n(n+l] distances between the n poles of the axes and the pole of the vector. We therefore call Yn the Surface Harmonic of order n.

130#.] We have next to shew that to every surface-harmonic of order n there corresponds not only a solid harmonic of degree — (n+ 1) but another of degree n, or that

js.= r,,i- = ^1f«-« (15)

satisfies Laplace's equation.

1 30 6.] SOLID HARMONIC OF POSITIVE DEGREE. 183

For

Hence

Now, since ^ is a homogeneous^ function of. ..0v...#,.jind 2, of negative degree n + 1~

r.. (17)

dz J '

The first two terms therefore of the right-hand member of equation (16) destroy each other, and, since Vn satisfies Laplace's equation, the third term is zero, so that //„ also satisfies LaplaceV equation, and is therefore a solid harmonic of degree n.

This is a particular case of the more general theorem of electrical inversion, which asserts that if F (%, y> z) is a function of a?, y, and z which satisfies Laplace's equation, then there exists another function, a a*x azy a2z

• 4 {— -$-> —;-) -Sj*

r \ r2 r2 r2 '

which also satisfies Laplace's equation. See Art. 162.

1303.] The surface harmonic Tn contains 2n arbitrary variables, for it is defined by the positions of its n poles on the sphere, and each of these is defined by two coordinates.

Hence the solid harmonics Vn and Hn also contain 2n arbitrary variables. Each of these quantities, however, when multiplied by a constant, will still satisfy Laplace's equation.

To prove that AHn is the most general rational homogeneous function of degree n which can satisfy Laplace's equation, we observe that K, the general rational homogeneous function of degree n, contains J («-f l)(»4-2) terms. But V2K is a homo- geneous function of degree w— 2, and therefore contains \n{n— -1) terms, and the condition V2K= 0 requires that each of these must vanish. There are therefore \n(n— 1) equations between the

184 SPHERICAL HARMONICS. [131 a.

coefficients of the \ (n+ 1) (n + 2) terms of the function K, leaving 2^ + 1 independent constants in the most general form of the homo- geneous function of degree n which satisfies Laplace's equation. But Hn, when multiplied by an arbitrary constant, satisfies the required conditions, and has 2 # + 1 arbitrary constants. It is therefore of the most general form.

131 aJ\ We are now able to form a distribution of potential such that neither the potential itself nor its first derivatives become infinite at any point.

The function Fn = Ynr~(n+l) satisfies the condition of vanishing at infinity, but becomes infinite at the origin.

The function Hn=Ynrn is finite and continuous at finite dis- tances from the origin, but does not vanish at an infinite distance.

But if we make anYnr~(n+l) the potential at all points outside a sphere whose centre is the origin, and whose radius is a, and «-(ri+1) Ynrn the potential at all points within the sphere, and if on the sphere itself we suppose electricity spread with a surface density cr such that

47T(T«2 = (2n+l)Ynt (18)

then all the conditions will be satisfied for the potential due to a shell charged in this manner.

For the potential is everywhere finite and continuous, and vanishes at an infinite distance ; its first derivatives are everywhere finite and are continuous except at the charged surface, where they satisfy the equation

• f: +4 „„ = <>, (19)

and Laplace's equation is satisfied at all points both inside and outside of the sphere.

This, therefore, is a distribution of potential which satisfies the conditions, and by Art. 100 a it is the only distribution which can satisfy them.

131 #.] The potential due to a sphere of radius a whose surface density is given by the equation

47Tfl2(7=:(2^+l)7n, (20)

is, at all points external to the sphere, identical with that due to the corresponding singular point of order n.

Let us now suppose that there is an electrical system which we may call U, external to the sphere, and that ^ is the potential due to this system, and let us find the value of 2 (^tf) for the

131 C.] SINGULAR POINT EQUIVALENT TO CHARGED SHELL. 185

singular point. This is the part of the electric energy depending on the action of the external system on the singular point.

If AQ is the charge of a single point of order zero, then the potential energy in question is

ȣ = 4>*. (21)

If there are two such points, a negative one at the origin and a positive one of equal numerical value at the extremity of the axis hl} then the potential energy will be

• 4 VJT + *«•'

and when AQ increases and h^ diminishes indefinitely, but so that AQ&! = A13 the value of the potential energy for a point of the first order will be

(22)

Similarly for a point of order n the potential energy will be

1 fJn j/

^ = — r — 4. 7l „ *• (23)

1.2. ..n n dh^...dhn

131 <?.] If we suppose the external system to be made up of parts, any one of which is denoted by dE> and the singular point to be made up of parts any one of which is de, then

(24) But if Vn is the potential due to the singular point,

5 = S(1&), (25)

and the potential energy due to the action offione is

= 22 (-dEde) = 2rndJ$, (26)

the last expression being the potential energy due to the action of e on E.

Similarly, if <rds is an element of electricity on the shell, since the potential due to the shell at the external system E is Fn, we have

W= 2(TndE) = 22(-4#»*)« 2(*cr<fc). (27)

The last term contains a summation to be extended over the

• We shall find it convenient, in what follows, to denote the product of the positive integral numbers 1.2. 3... w^byw^ ^f

186 SPHERICAL HARMONICS. [132.

surface of the sphere. Equating it to the first expression for Wt we have

If we remember that 4 K a- a2 = (2 n+ 1) Zw, and that Jn = flw, this becomes

7 ^ = _ - ^«+2 . (29)

ra! (2»+l) dh^...d/in

This equation reduces the operation of taking the surface integral of ^ Tn ds over every element of the surface of a sphere of radius a, to that of differentiating ^ with respect to the n axes of the harmonic and taking the value of the differential coefficient at the centre of the sphere, provided that ty satisfies Laplace's equa- tion at all points within the sphere, and Yn is a surface harmonic of order n.

132.] Let us now suppose that ^ is a solid harmonic of positive degree m of the form

V = a-mYmrm. (30)

At the spherical surface, r = a, and ^ = Ym, so that equation (29) becomes in this case

ft! (2^+1) dh^...dhn

where the value of the differential coefficient is to be taken at the centre of the sphere.

When n is less than m, the result of the differentiation is a homogeneous function of so, y and z of degree m—n, the value of which at the centre of the sphere is zero. If n is equal to m the result of the differentiation is a constant, the value of which we shall determine in Art. 134 6. If the differentiation is carried

further, the result is zero. Hence the surface-integral YmYn ds

vanishes whenever m and n are different.

The steps by which we have arrived at this result are all of them purely mathematical, for though we have made use of terms having a physical meaning, such as electrical energy, each of these terms is regarded not as a physical phenomenon to be investigated, but as a definite mathematical expression. A mathematician has as much right to make use of these as of any other mathematical functions which he may find useful, and a physicist, when he has

1 33.] TRIGONOMETRICAL EXPRESSION. 187*

to follow a mathematical calculation, will understand it all the better if each of the steps of the calculation admits of a physical interpretation.

133.] We shall now determine the form of the surface harmonic Yn as a function of the position of a point P on the sphere with respect to the n poles of the harmonic.

We have

and so on.

Every term of Yn therefore consists of products of cosines, those of the form /u, with a single suffix, being- cosines of the angles between P and the different poles, and those of the form A, with double suffixes, being cosines of the angles between the poles.

Since each axis is introduced by one of the n differentiations, the symbol of that axis must occur once and only once among the suffixes of the cosines of each term.

Hence if in any term there are s cosines with double suffixes, there must be n—2 s cosines with single suffixes.

Let the sum of all products of cosines in which s of them have double suffixes be written in the abbreviated form

S(|un-28A8).

In every one of the products all the suffixes occur once, and none is repeated.

If we wish to express that a particular suffix, m, occurs among the /u's only or among the A's only, we write it as a suffix to the \L or the A. Thus the equation

2(M»-*«A<) = 2(Mm»-^A') + 2(M"-2'A,,.') (33)

expresses that the whole set of products may be divided into two parts, in one of which the suffix m occurs among the direction cosines of the variable point P, and in the other among the cosines of the angles between the poles.

Let us now assume that for a particular value of n

820*«-2«A«) + &c., (34)

when the A's are numerical coefficients. We may write the series in the abbreviated form

rn = S[A.82(Mw-28A*)], (35)

when S indicates a summation in which all values of a, including zero, not greater than \n^ are to be taken.

188 SPHERICAL HARMONICS. [133.

To obtain the corresponding solid harmonic of negative degree (rc+ 1) and order n, we multiply by /-(n+1), and obtain

r. = 8\A... r*"»-i 2 (j»-»- V)] ; (36)

putting rfj. = p, as in equation (3).

If we differentiate Vn with respect to a new axis km we obtain and therefore

-An.sr2'-*n-lI,(pn-2°-l.Xms+l)]. (37)

If we wish to obtain the terms containing s cosines with double suffixes, we must diminish s by unity in the last term, and we find

V)

]- (38)

Now the two classes of products are not distinguished from each other in any way except that the suffix m occurs among the jt/s in one and amoug the A's in the other. Hence their coefficients must be the same, and since we ought to be able to obtain the same result by putting n -t- 1 for n in the expression for Vn and multiplying by n--l, we obtain the following equations,

(n+l)An+li8 = (2»-2«-f 1)4... = -4.,.-!. (39)

If we put s = 0, we obtain

(n+l)An+L = (2n+l)An'} (40)

and therefore, since AltQ = 1,

2tt! , ,

A'°-2^)2; and from this we obtain the general value of the coefficient

(2^-2.)! . ^.s-(-J 2n-.wI(tt-Jj)I»

and finally the trigonometrical expression for the surface harmonic, as

y.-^K-y'^"^- • (43)

This expression gives the value of the surface harmonic at any point P of the spherical surface in terms of the cosines of the distances of P from the different poles and of the distances of the poles from each other.

It is easy to see that if any one of the poles be removed to the opposite point of the spherical surface, the value of the harmonic will have its sign reversed. For any cosine involving

134.] //XYtlck 189

the index of this pole will have its sign reversed, and in each term of the harmonic the index of the pole occurs once and only once.

Hence if two or any even number of poles are removed to the points respectively opposite to them, the value of the harmonic will be unaltered.

Professor Sylvester, however, has shewn (Phil. Mag., Oct. 1876) that when the harmonic is given, the problem of finding the n lines which coincide with the axes has one and only one solution, • though, as we have just seen, the directions to be reckoned positive along these axes may be reversed in pairs.

134.] We are now able to determine the value of the surface

Yn ds when the order of the two surface harmonics

integral / /

is the same, though the directions of their axes may be in general different.

For this purpose we have to form the solid harmonic Ymrn and to differentiate it with respect to each of the n axes of Yn .

Any term of Ymr£ of the form rm^m-28X.8 may be written r2spmm~28Xmm8. Differentiating this n times in succession with respect to the n axes of Yn, we find that in differentiating r28 with respect to s of these axes we introduce s of the jon's, and

the numerical factor

2s (2s— 2). ..2, or 2*s \

In continuing the differentiation with respect to the next s axes, the jt?n's become converted into Xnn's, but no numerical factor is introduced, and in differentiating with respect to the remaining n — 2s axes, the pm's become converted into Awn's, so that the result is 2*sl\nn8\mms\mnm-28.

We have therefore, by equation (31),

and by equation (43),

Hence performing the differentiations and remembering that m = n, we find

a? of, v(2ft-2*)''g/x. x- A"-' n. US') 1«!25L(~) 2-»(.-.)!S(X«"X"X >\

,« X/.. / j #*tA J. «•**"'

190 SPHERICAL HARMONICS.

135 aJ\ The expression (46) for the surface-integral of the product of two surface-harmonics assumes a remarkable form if we suppose all the axes of one of the harmonics, Ym, to coincide with each other, so that Ym becomes what we shall afterwards define as the zonal harmonic of order m, denoted by the symbol Pm .

In this case all the cosines of the form \nm may be written JUH, where pn denotes the cosine of the angle between the common axis of Pm and one of the axes of Tn. The cosines of the form \mm will all become equal to unity, so that for 2Asmm we must put the number of combinations of s symbols, each of which is distinguished by two suffixes out of n, no suffix being repeated. Hence

The number of permutations of the remaining n—2s indices of the axes of Pm is (n — 2 s) ! Hence

2 (^~2s) = (n—2s) ! juw-28. (48)

Equation (46) therefore becomes, when all the axes of Ym coincide

with each other,

£ ;:-..-.

A 2

), ty equation (43), (50)

2n+l where Yn (m) denotes the value of Yn at the pole of Pm .

We may obtain the same result by the following shorter pro- cess : —

Let a system of rectangular coordinates be taken so that the axis of z coincides with the axis of Pm) and let Ynrn be expanded as a homogeneous function of #, y> z of degree n.

At the pole of Pm , x = y = 0 and z = r, so that if Czn is the term not involving x or y, C is the value of Yn at the pole of Pm.

Equation (31) becomes in this case

If m is equal to n, the result of differentiating Czn is n\ C, and is zero for the other terms. Hence

C being the value of Yn at the pole of Pm.

135 b.] This result is- a very important one in the theory of

I35&-] EXPANSION IN SPHERICAL HARMONICS. 191

spherical harmonics, as it shews how to determine a series of spherical harmonics which expresses the value of a quantity having any arbitrarily assigned finite and continuous value at each point of a spherical surface.

For let F be the value of the quantity and ds the element of surface at a point Q of the spherical surface, then if we multiply Ids by Pn, the zonal harmonic whose pole is the point P of the same surface, and integrate over the surface, the result, since it depends on the position of the point P, may be considered as a function of the position of P.

But since the value at P of the zonal harmonic whose pole is Q is equal to the value at Q of the zonal harmonic of the same order whose pole is P, we may suppose that for every element ds of the surface a zonal harmonic is constructed having its pole at Q and having a coefficient Fds.

We shall thus have a system of zonal harmonics superposed on each other with their poles at every point of the sphere where F has a value. Since each of these is a multiple of a surface harmonic of order n, their sum is a multiple of a surface harmonic (not necessarily zonal) of order n.

The surface integral / / FPnds considered as a function of the point P is therefore a multiple of a surface harmonic Yn ; so that

is also that particular surface harmonic of the nih order which belongs to the series of harmonics which expresses F, provided F can be so expressed.

For if F can be expressed in the form

then if we multiply by Pnds and take the surface integral over the whole sphere, all terms involving products of harmonics of different orders will vanish, leaving

Hence the only possible expansion of F in spherical harmonics is

(51)

192 SPHEEICAL HARMONICS. [137.

Conjugate Harmonics.

136.] We have seen that the surface integral of the product of two harmonics of different orders is always zero. But even when the two harmonics are of the same order, the surface integral of their product may be zero. The two harmonics are then said to be conjugate to each other. The condition of two harmonics of the same order being conjugate to each other is expressed in terms of equation (46) by making its members equal to zero.

If one of the harmonics is zonal, the condition of conjugacy is that the value of the other harmonic at the pole of the zonal harmonic must be zero.

If we begin with a given harmonic of the nih order, then, in order that a second harmonic may be conjugate to it, its 2n variables must satisfy one condition.

If a third harmonic is to be conjugate to both, its 2 n variables must satisfy two conditions. If we go on constructing harmonics, each of which is conjugate to all those before it, the number of conditions for each will be equal to the number of harmonics already in existence, so that the (2^+l)th harmonic will have 2n conditions to satisfy by means of its 2 n variables, and will therefore be completely determined.

Any multiple A Tn of a surface harmonic of the nih order can be expressed as the sum of multiples of any set of 2 n + 1 conjugate harmonics of the same order, for the coefficients of the 2^+1 conjugate harmonics are a set of disposable quantities equal in number to the 2 n variables of Yn and the coefficient A.

In order to find the coefficient of any one of the conjugate harmonics, say Yn*, suppose that

Multiply by Yn*ds and find the surface integral over the sphere. All the terms involving products of harmonics conjugate to each other will vanish, leaving

AJJT. rn'ds = 4,//(r/)" 4, (52)

an equation which determines A0.

Hence if we suppose a set of 2ti+l conjugate harmonics given, any other harmonic of the nih order can be expressed in terms of them, and this only in one way. Hence no other harmonic can be conjugate to all of them.

137.] We have seen that if a complete system of 2#+l har-

138.] ZONAL HARMONICS. 193

monies of the nih order, all conjugate to each other, be given, any other harmonic of that order can be expressed in terms of these. In such a system of 2 n -{- 1 harmonics there are 2n(2n+l) variables connected by n(2n+l) equations, n(2n+l) of the variables may therefore be regarded as arbitrary.

We might, as Thomson and Tait have suggested, select as a system of conjugate harmonics one in which each harmonic has its n poles distributed so that j of them coincide at the pole of the axis of x, Jc at the pole of ^, and l(= n—j—Jc) at the pole of z. The n -f 1 distributions for which I = 0 and the n distributions for which 1=1 being given, all the others may be expressed in terms of these.

The system which has been actually adopted by all mathe- maticians (including Thomson and Tait) is that in which n— a- of the poles are made to coincide at a point which we may call the Positive Pole of the sphere, and the remaining a poles are placed at equal distances round the equator when their number is odd, or at equal distances round one half of the equator when their number is even.

In this case ju1} /u2, . . . /txn_a are each of them equal to cos 0, which we shall denote by /u. If we also write v for sin 0, /otn_<r+1 .../un are of the form v cos ($— /3), where /3 is the azimuth of one of the poles on the equator.

Also the value of \pq is unity, if p and q are both less than n— <r, zero when one is greater and the other less than this number, and cos r TT/J when both are greater, r being an integral number less than a-.

138.] When all the poles coincide at the pole of the sphere, (7 = 0, and the harmonic is called a Zonal harmonic. As the zonal harmonic is of great importance we shall reserve for it the symbol Pn.

We may obtain its value either from the trigonometrical ex- pression (43) or more directly by differentiation, thus

_ 1.3.5...(2»— 1 n~~ 1.2.3... n I/" 2.(2« —

(72J--T 2^3 ^'"4 H

^fxn-2pl, (53)

Lx ' z"^? i \n—p) i ^w — 2/?) 1

VOL. I. o

194: SPHERICAL HARMQNICS. [139.

where we must give to p every integral value from zero to the greatest integer which does not exceed \n.

It is sometimes convenient to express Pn as a homogeneous function of cos 0 and sin 0, or, as we write them, ju and v,

{54)

It is shewn in the mathematical treatises on this subject that Pn(\j) is the coefficient of hn in the expansion of (1 — 2ju^-f 7i?)-%. The surface integral of the square of the zonal harmonic, or

+i 2

(55)

Hence f" (P. (M))2 «*M = ^ • (56)

139.] If we consider a zonal harmonic simply as a function of /u, and without any explicit reference to a spherical surface, it may be called a Legendre's Coefficient.

If we consider the zonal harmonic as existing on a spherical surface the points of which are defined by the coordinates 0 and <£, and if we suppose the pole of the zonal harmonic to be at the point (0', </>'), then the value of the zonal harmonic at the point (0, $) is a function of the four angles 0', $', 0, $, and because it is a function of jut, the cosine of the arc joining the points (0, </>) and (0', <£'), ii> will be unchanged in value if 0 and 0', and also <£ and $', are made to change places. The zonal harmonic so expressed has been called Laplace's Coefficient. Thomson and Tait call it the Biaxal Harmonic.

Any homogeneous function of as, y, z- which satisfies Laplace's equa- tion may be called a Solid harmonic, and the value of a solid harmonic at the surface of a sphere whose centre is the origin may be called a Surface harmonic* In this book we have defined a surface harmonic by means of its n poles, so that it has only 2n variables. The more general surface harmonic, which has 2n--l variables, is the more restricted surface harmonic multiplied by an arbitrary constant. The more general surface harmonic, when expressed in terms of 0 and $, is called a Laplace's Function.

140 0.] To obtain the other harmonics of the symmetrical system, we have to differentiate with respect to <r axes in the plane of xy inclined to each other at angles equal to TT/O; This may be most

140 «•] TESSERAL HARMONICS. 195

conveniently done by means of the system of imaginary coordinates given in Thomson and Tait's Natural Philosophy, vol. I, p. 148 (or p. 185 of 2nd edition). If we write

where i denotes */ — 1, the operation of differentiating with respect to the o- axes may be written

-<

if one of the axes coincides with the axis of v^and

if the axis of y bisects the angle between two of the axes/" " ' ' We shall find it convenient to express these operations by the

abbreviated symbols of operation Ds and Dc^ respectively. They are, of course, real operations, and may be expressed without the use of imaginary symbols thus —

(<r) d*~l d or (a- — l}((r — 2^ d*~z dz

off— 1 7) o — «• _^ IS L

|fec H

1.2

We shall also write

" r><7 ~ 7^ n

-= - Ds = Ds and -j—-Dc = Dc't (62)

^n-°- n ^"-^ »'

(») W

so that i><9 and DC denote the operations of differentiating with

n n

respect to n axes, n — <r of which coincide with the axis of 2, while the remaining o- make angles TT/O- with. each other in the plane of

(•)

#y, D* being used when the axis of y coincides with one of the

»

(<r)

axes, and DC when the axis of y bisects the angle between two

n

of the axes.

The two tesseral surface harmonics of order n and type o- may now be written

• (G4)

O 2

196 SPHERICAL HARMONICS. [140 a.",

Writing /u, = cos 0, v = sin 0, p2 = as2 -f-y2, so that 2 = fir, p = vr, x = p cos <f>, y = p sin $,

Behave I/? I = (-ifM^-^-Ly, (65)

^H-rg^^. <««>

in which we may write

I Or-f*) = P" sin <r& i (£* + iT) = f cos cr<#>. (67)

We have now only to differentiate with respect to z, which we may do either so as to obtain the result in terms of r and z, or as a homogeneous function of z and p divided by a power of r,

(2«r)l

If we write

.

and

(n—<r)(n — <r— 1)(^ — o-— 2)f^— o-— 3) "1

• • 2.4.(2;-1)(2a-3)    -  ^""^  -  &c-]  ' 
    

so that these two functions differ only by a constant factor.

We may now write the expressions for the two tesseral harmonics of order n and type cr in terms either of 0 or ^,

(74)

We must remember that when o- = 0, sin o-<^ = 0 and cos o-<£ = 1.

140 C. TESSERAL HARMONICS. 197

For every value of a- from 1 to n inclusive there is a pair of

(0) (0)

harmonics, but when o- = 0, Ys = 0 and Yc = 2PM, the zonal har-

« n

monic. The whole number of harmonics of order n is therefore 2u+ 1, as it ought to be.

140 £.] The numerical value of Y adopted in this treatise is that which we find by differentiating r'1 with respect to the n axes and dividing by n \ It is the product of four factors, the sine or cosine of 0-0, r0", a function of /x (or of //, and v), and a numerical co- efficient.

The product of the second and third factors, that is to say, the part depending on 0, has been expressed in terms of three different symbols which differ from each other only by their numerical factors. When it is expressed as the product of v* into a series of descending powers of ^ the first term being /j,n~°, it is the function which we, following Thomson and Tait, denote by 0.

The function which Heine (Handbuch der Kugelfunctionen, §47) denotes by P^ and calls eine zugeordnete Function erster Art> or, as Todhunter translates it, an ' Associated Function of the First Kind,' is related to ©^ by the equation

ej?=(-)fpj?. (75)

The series of descending powers of /x, beginning with /x"-*7, is expressed by Heine by the symbol ^\ and by Todhunter by the symbol ar(o^ n).

This series may also be expressed in two other forms,

n—(r\ dn+<r f „ Xn (^l)

2«(«-«r)!»l d*

*'

The last of these, in which the series is obtained by differentiating the zonal harmonic with respect to jx, seems to have suggested the symbol adopted by Ferrers, who defines it thus

7**- «* d* P- (2n^' e(<r)- (77)

ln • ' -d^°n-2«(n-«)\nl.n

When the same quantity is expressed as a homogeneous function of p and v, and divided by the coefficient of /x"-0" v°, it is what we have already denoted by yp.

140 c.] The harmonics of the symmetrical system have been classified by Thomson and Tait with reference to the form of the spherical curves at which they become zero.

198 SPHEEICAL HARMONICS. [141.

The value of the zonal harmonic at any point of the sphere is a function of the cosine of the polar distance, which if equated to zero gives an equation of the nib degree, all whose roots lie between — 1 and -f 1 , and therefore correspond to n parallels of latitude on the sphere.

The zones included between these parallels are alternately positive and negative, the circle surrounding the pole being always positive.

The zonal harmonic is therefore suitable for expressing a function which becomes zero at certain parallels of latitude on the sphere, or at certain conical surfaces in space.

The other harmonics of the symmetrical system occur in pairs, one involving the cosine and the other the sine of cr<£. They therefore become zero at a- meridian circles on the sphere and also at n — (T parallels of latitude, so that the spherical surface is divided into 2o- (n—a—1) quadrilaterals or tesserae, together with 4<r triangles at the poles. They are therefore useful in investigations relating to quadrilaterals or tesserae on the sphere bounded by meridian circles and parallels of latitude.

They are all called Tesseral harmonics except the last pair, which becomes zero at n meridian circles only, which divide the spherical surface into 2n sectors. This pair are therefore called Sectorial harmonics.

141.] We have next to find the surface integral of the square of any tesseral harmonic taken over the sphere. This we may do by the method of Art. 134. We convert the surface harmonic T^ into a solid harmonic of positive degree by multiplying it by rn, we differentiate this solid harmonic with respect to the n axes of the harmonic itself, and then make x — y = z = 0, and we multiply the

., ,

result by —

J n\

These operations are indicated in our notation by

). (78)

Writing the solid harmonic in the form of a homogeneous func- tion of z and f, 77, viz.,

rnls =

we find that on performing the differentiations with respect to z, all the terms of the series except the first disappear, and the factor (# — <r)I is introduced.

T42 &•] SURFACE INTEGRALS. 199

Continuing the differentiation with respect to f and T? we get rid also of these variables and introduce the factor <rl, so that the final result is

.

2a*»!»!

We shall denote the second member of this equation by the abbreviated symbol [n, a].

This expression is correct for all values of <r from 1 to n inclusive, but there is no harmonic in sin 0-$ corresponding to <r = 0.

In the same way we can shew that

2n+l 2 for all values of a- from I to n inclusive.

When o- = 0, the harmonic becomes the zonal harmonic, and

a result which may be obtained directly from equation (50) by putting 7n = Pm and remembering that the value of the zonal harmonic at its pole is unity.

14,2 a.] We can now apply the method of Art. 136 to determine the coefficient of any given tesseral surface harmonic in the expansion of any arbitrary function of the position of a point on a sphere. For let F be the arbitrary function, and let A°n be the coefficient of Y^' in the expansion of this function in surface harmonics of the symmetrical system

)'* = 4* ["»"]. (83)

where [n, cr] is the abbreviation for the value of the surface integral given in equation (80).

142 $.] Let ty be any function which satisfies Laplace's equation, and which has no singular values within a distance a of a point 0, which we may take as the origin of coordinates. It is always possible to expand such a function in a series of solid harmonics of positive degree, having their origin at 0.

One way of doing this is to describe a sphere about 0 as centre with a radius less than a, and to expand the value of the potential at the surface of the sphere in a series of surface harmonics. Multiplying each of these harmonics by r/a raised to a power equal to the order of the surface harmonic, we obtain the solid harmonics of which the given function is the sum.

200 SPHERICAL HARMONICS. [T43-

But a more convenient method, and one which does not involve integration, is by differentation with respect to the axes of the harmonics of the symmetrical system.

For instance, let us suppose that in the expression of ^, there is

00 (<0 a term of the form Ac Yc rn.

n n

If we perform on ^ and on its expansion the operation dn-°- ,d* d* ^ d^^^d^ * *?'' and put x, y, z equal to zero after differentiating1, all the terms