And, if σ, be the corresponding surface density,

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 21+1 arbitrary constants.

For the solid spherical harmonic, Y, being a rational integral function of x, y, z of the ith degree, will consist of (i + 1) (i + 2) terms. Now the expression VV, being a rational integral function of x, y, z of the degree ¿ – 2, will and the condition that it must be

2

consist of

=

(i − 1) i
2

terms;

0 for all values of x, y, z, will give rise to

coefficients of these terms, leaving

or 2i+1, independent coefficients.

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 r*P ̧ (μ), is a function of z and r of the degree i, d2 V d2 V ď2 V satisfying the equation V2V=0, or + + dx2 dy

dz2

= 0.

Now, if we denote this expressión by P, (z), we see that

since it is a function of z and r, it is a function of the distance (2) 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 d2V ď2 V leaving r unaltered, the équation + + =0 will dx2 + dy2 dz2 continue to be satisfied.

Now z+a (x+√1y), 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+√1y)} is a solid zonal harmonic of the degree i, its axis being the imaginary line

=

α a√-1

= 2.

is satisfied by V=P1 {z + a (x + √—1y)}, that is, expanding by Taylor's Theorem, it is satisfied by

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,

(x+√=1y)dP,(z), (x+√=1y}zdaP,(z)

dz

satisfies the equation 0.

dz2

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

(x−√=1y) dP,(z), (x−√—1y)2 d1P, (*), ... (x−√—1y) d'P, (2),

dz

satisfies the same equation.

dz2

dzi

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 √-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

6. We may deduce the surface harmonics from these by writing r sin cos o for x, r sin 0 sin o for y, r cos 0 for z, and dividing by r. Then, putting cosμ, and observing dP,(z) dP(u) that P(x)=rP¿(μ),

lowing series of 2+1 solutions:

we obtain the fol

=pt

dz

...

αμ

(C and S denoting any quantities independent of 0 and 4) are called Tesseral Surface Harmonics of the degree i and order σ. The particular forms assumed by them when σ =¿ are called Sectorial Surface Harmonics of the degree i. It will be observed that, since is a numerical constant,

d'P¿(μ)
αμ

Sectorial Harmonics only involve in the form

The product obtained by multiplying a Tesseral or Sectorial Surface Harmonic of the degree i by * (that is, the expression directly obtained in Art. 5) is called a Tesseral or Sectorial Solid Harmonic of the degree i

7. We shall denote the factor of a Tesseral or Sectorial

Harmonic which is a function of 0, that is sin 0

αμσ

,

or

(1 — μ2) doo, (μ), by the symbol T("), or, when it is necessary

(エーパ)

αμσ

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

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 in a completely developed form.

dito

i

24

i

duito

1

1.2

με

Now do {4+ (-1) -...}.

= 2i (2i−1)...(i - σ +1) μi-o

= 2i (2i — 1)...(îi — o + 1) {μ¿ê▬• _ (i — o) (i — o — 1)

+

2.2-1

-1) -

ཙཧཾ ...}.

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

A being a quantity independent of 0 and 4. The factor of this, involving μ, is denoted by Thomson and Tait (Natural Philosophy, Vol. 1, p. 149) by the symbol . Thomson and Tait also employ a symbol 9,, adopted by Maxwell in his Treatise on Electricity and Magnetism, Vol. 1, p. 164, which is equal to