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And, if o, be the corresponding surface density,

2i +1 0;=


4c It hence follows that if the potential of a spherical shell, of indefinitely small thickness, be a surface harmonic, its potential at any internal point will be proportional to the corresponding solid harmonic of positive degree, and its potential at any external point will be proportional to the corresponding solid harmonic of negative degree.

That is, the proposition proved for zonal harmonics in Chap. III. Art. 6, is now extended to spherical harmonics in general.

4. The spherical harmonic of the degree i will involve 2i+1 arbitrary constants.

For the solid spherical harmonic, 7oY;, being a rational integral function of x, y, z of the żth degree, will consist of (i + 1) (i + 2)

terms. Now the expression VV, being a 2 rational integral function of x, y, z of the degree i - 2, will consist of

(i – 1) i

terms; and the condition that it must be 2

(i-1) i O for all values of x, y, z, will give rise to


2 (8 + 1) (+ 2) coefficients of these terms, leaving

2 (i+1) (+ 2) (i – 1) i

, or 2i +1, independent coefficients. 2



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5. We proceed to shew how the spherical harmonic of the degree i may be arranged in a series of terms, each of which may be deduced by differentiation' from the Zonal Harmonic symmetrical about the axis of z. The solid zonal harmonic, which, in accordance with the notation already employed, is represented by riP;(), is a function of x and r of the degree i,

dy do V MV

V satisfying the equation VR V= 0, or +

= 0. da* * ay * da

dy? Now, if we denote this expression by P: (z), we see that





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= 2.


since it is a function of % and r, it is a function of the distance (z) from a certain plane passing through the origin, and of the distance (r) from the origin. Further, if we write for z the distance from any other plane passing through the origin,

d? V d V da V leaving r unaltered, the equation + + =0 will

dx2 dy2

dza continue to be satisfied:

Now 2+2(x+1=1 y), a being any quantity whatever, represents the distance from a certain plane passing through the origin, since in this expression, the sum of the squares of the coefficients of z; x, y is equal to unity. Hence P: {z +a (x+7-1y)} is a solid zonal harmonic of the

y degree i, its axis being the imaginary line

an-1 Therefore the equation

d? V dal da V
de + dy + da

= 0,

: dz? is satisfied by V=P,{z+a (ac+V – 1 y)}, that is, expanding by Taylor's Theorem, it is satisfied by

dP; (2)

a? P: (z) + (x+V-1y) + (x+v=1y): d®P, (2) +....


dza ta" (x+V 1 y)* d'P, (2)

a V

1.2... dzi for all values of a.

Hence, since the equation in V is linear, it follows that it is satisfied by each term separately, or that, besides P. (2) itself, each of the i expressions,

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dza satisfies the equation V=0.

By similar reasoning we may shew that each of the i expressions, dP

( )


dzi satisfies the same equation.

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Now each of the 2i solutions, thus obtained, is imaginary. But the sum of any two or more of them, or the result obtained by multiplying any two or more by any arbitrary quantities, and adding the results together, will also be a solution of the equation. Hence, adding each term of the first series to the corresponding term of the second, we obtain a series of i real solutions of the equation. Another such series may be obtained by subtracting each term of the second series from the corresponding term of the first, and dividing by N-1. We have thus obtained (including the original term P.(z)) a series of 2i +1 independent solutions of the given equation, which will be the 2i + 1 independent solid harmonics of the degree i.

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.... cos ipsin'o d'P. (2)

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6. We may deduce the surface harmonics from these by writing r sin 8 cos 0 for x, r sin sin $ for y, r cos 0 for x, and dividing by m. Then, putting cos O=x, and observing


dP (2) that P: ()=r'P,(M),


we obtain the foldz

du lowing series of 2i +1 solutions :

Pi(u), dP;()

d’P:(u) cos sin 0 cos 26 sino du du

du dP, (u), sin 2$sin’q d*P;()

d'P, (e) sin o sino


... sin id sin'e

du Expressions of the form

doPi(u) Ccoς σφ since


doP:(u) S sin op sino

duo or their equivalents,

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doPi (u) Ssin o$ (1 - re)


order o.

(C and S denoting any quantities independent of O and ) are called Tesseral Surface Harmonics of the degree i and

The particular forms assumed by them when ori are called Sectorial Surface Harmonics of the degree i. It will be observed that, since


is a numerical constant,

Sectorial Harmonics only involve 0 in the form

sin“0, or (1 –re).
The product obtained by multiplying a Tesseral or
Sectorial Surface Harmonic of the degree i by got (that is,
the expression directly obtained in Art. 5) is called a Tesseral
or Sectorial Solid Harmonic of the degree i.





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7. We shall denote the factor of a Tesseral or Sectorial Harmonic which is a function of e, that is sin odoP.(),


duo (1-1) i do P. (k), by the symbol T), or, when it is necessary u

( duo to particularize the quantity of which it is a function, by T. 6) (u) or T.(o) (cos 6).

It will be convenient, for the purpose of comparison with the forms of Tesseral Harmonies given in the Mécanique Céleste, and elsewhere, to obtain T/C) in a completely developed form.

1 d' (u? — 1)' Now, sinee P:(u)

we see that i du doP;(u)

1 di+o (m-1) duo i

duito 1 dito


jis i duito


i(i-1) _...}.

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dito ,


= 2i (2i – 1)...(i – 0+1) Hi-o

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i(8 – 1)

(2i – 4) (21 – 5)...(i-0-3-0-4

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= 2i (21 – 1)...(–o+1) {us-o

(i - 0) (-0-1) (i0) (0-0-1) (i=0=2) (0-0-5) wi-----...}

. i



2.4.(2i – 1) (2i 3)

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And therefore

(i-o)(-o-1) TO)=


) i2-0-1)(0-2()

2.4 (21-1) (21—3) The form given by Laplace for a Tesseral Surface Harmonic of the degree i and order o is (see Mécanique Céleste, Liv. 3, Chap. 2, pp. 40–47)

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() ( -0-1

COS 00, 2 (2i – 1) A being a quantity independent of 0 and $. The factor of

0 this, involving M, is denoted by Thomson and Tait (Natural Philosophy, Vol. 1, p. 149) by the symbol 0,0). Thomson and Tait also employ a symbol giro), adopted by Maxwell in his Treatise on Electricity and Magnetism, Vol. 1, p. 164, which is equal to 1.2...o

(i + o) (ito-1)...i -0+1).


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1.2...o 20

T.CO). (i +o) (i to -1)...(i-o + 1)

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