whence, if mi be not = 0,

The value of

after.

[`[Y;dud will be investigated here

11. We may hence prove that if a function of μ and o can be developed in a series of surface harmonics, such development is possible in only one way.

For suppose, if possible, that there are two such developments, so that

and also

F (μμ, $) = Y2+ Y2+ ... + Y1 + ...

F (μ, $) = Y,' + Y1' + ... + Y + ...

Then subtracting, we have

0=Y-Y'+Y12- Y,' + ... + Y - Y+... identically,

1

1

Now, each of the expressions Y.-Y, Y.-Y... Y-Y being the difference of two surface harmonics of the degree 0, 1, ... ¿... is_itself a surface harmonic of the degree 0, 1, ....... Denote these expressions for shortness by Z。, Z... Z... so that

0=Z2+Z2+...+Z+... identically.

Then, multiplying by Z, and integrating all over the surface of the sphere, we have

=

That is, the sum of an infinite number of essentially positive quantities is 0. This can only take place when each of the quantities is separately = 0. Hence Z, is identically =0, or Y = Y, and therefore the two developments are identical.

We have not assumed here that such a development is always possible. That it is so, will be shewn hereafter.

12. By referring to the expression for a surface harmonic given in Art. 4, we see that each of the Tesseral and Sectorial Harmonics involves (1-2), or some power of (1 - μ2), as a factor, and therefore is equal to 0 when μ=±1. From this it follows that when μ=±1, the value of the Surface Harmonic is independent of 4, or that if Y (u, 4) represent a general surface harmonic, Y(+1, 4) is independent of , and may therefore be written as Y(+1). Or Y(1) is the value of Yu, 4) at the pole of the zonal harmonic P¡ (μ), Y(-1) at the other extremity of the axis of P. (u).

We may now prove that

2π

[** Y ̧dp = 2π Y, (1) P. (4).

For, recurring to the fundamental equation,

Now, if we integrate this equation with respect to 8, between the limits 0 and 27, we see that, since

and the value of Y, only involves & under the form of cosines or sines of and its multiples, and therefore the values of

dY:

аф

are the same at both limits, it follows that

~ 2π d2 Y;
'dp=0.
dp

Hence

d/ { (1 − μ3) (* 1,ap)} + i (i + 1) ([* Y,do) = 0.

άμ

аф

Hence Yap is a function of μ which satisfies the

fundamental equation for a zonal harmonic, and we therefore have

2π

[** Y1dp = CP,(u),

C being a constant, as yet unknown.

To determine C, put μ=1, then by the remark just made, Y, becomes Y(1), and is independent of p. Hence, when

Y,dp=2πY, (1). Also P.(u)=1. We have there

13. We may now enquire what will be the value of

Y, Z, being two general surface harmonics of the degree i Suppose each to be arranged in a series consisting of the zonal harmonic P, whose axis is the axis of z, and the system of tesseral and sectorial harmonics deduced from it. Let us represent them as follows:

AP

+CT cos + CT(2) cos 24+ ... + CT() cos σ$ +

&+

...

+ CT cos ip

+ ST sin &+ ST(2) sin 24 + ... + ST ̧ sin of +

aP

...

+ ST sin ip;

+c1T cos +c2T(2) cos 24 + ... + c.T.) cos op + ...

c.T() cos ip

+8,Tsin &+s,T(2) sin 24+...+s ̧Ã ̧‹°) sin σ$+...

+8,T.↔ sin ip.

Hence the product YZ, will consist of a series of terms, in which will enter under the form cos o cos o'o, or cos op sin o'p. This expression when integrated between

=

the limits 0 and 2π vanishes in all cases, except when oσ and the expression consequently becomes equal to cos2 σp, or sin2 σp. In these cases we know that, ☛ being any positive integer,

Hence the question is reduced to the determination of the value of

But, by the theorem of Rodrigues, proved in Chap. II. Art. 8, we know that

Now, putting (u2 — 1)* = M for the moment, and inte

grating by parts,

by a repetition of the same process.

And by repeating this process σ times, we see that

It will be observed that this result does not hold when σ = 0, in which case we have