since this is the form which equation (2) assumes when V is an homogeneous function of the degree i, Now, put V1 = gubiti U,, and this becomes 1 d

1 d i(i+1)p2öt? Uit pozit1 sin +

(**U)=0, sin* 0 do*

1

= i

(* o

2

sin de

do

or

1 d dU 1 đU, i(i+1) U. +

+

= 0 ..... (2). do

sin" e dos Now, since U, is a homogeneous function of the degree -(i + 1),

d (rU!) dU

= U, +r

;
dr

dr
=- iU;
d” (rU:)

dU
dr2

dr

U. i)

= -2

=i (i+1)

r

or

=

d' (rU)

=i (i+1) U::

dr2 therefore equation (2) becomes

1 d du 1 d°U
+
sin o
+

:0, ar sin Ꮎ dᎾ de sino do shewing that U, is an admissible value of V, as satisfying equation (2).

It appears therefore that every form of U, can be obtained from Vi, by dividing by päiti, and conversely, that every form of V, can be obtained from U, by multiplying by it? Such an expression as Vi we shall call a Solid Spherical Harmonic of the degree i. The result obtained by dividing Vi by 71, which will be a function of two independent variables 0 and $ only, we shall call a Surface Spherical Harmonic of the same degree. A very important form of spherical barmonics is that which is independent of $. The solid harmonics of this form will involve two of the variables, x and y, only in the form a*+y", or will be functions of x* + y and z. Harmonics independent of $ are called Zonal Harmonics, and are distinguished, like spherical harmonics in general, into Solid and Surface Harmonics. The investigation of their properties will be the subject of the following chapter.

The name of Spherical Harmonics was first applied to these functions by Sir W. Thomson and Professor Tait, in their Treatise on Natural Philosophy. The name “Laplace's Coefficients” was employed by Whewell, on account of Laplace having discussed their properties, and employed them largely in the Mécanique Céleste. Pratt, in his Treatise on the Figure of the Earth, limits the name of Laplace’s Coefficients to Zonal Harmonics, and designates all other spherical harmonics by the name of Laplace's Functions. The Zonal Harmonic in the case which we shall consider in the following chapter, i.e., in which the system is symmetrical about the line from which is measured, was really, however, first introduced by Legendre, although the properties of spherical harmonics in general were first discussed by Laplace; and Mr Todhunter, in his Treatise, on this account calls them by the name of “Legendre's Coefficients,” applying the name of “ Laplace's Coefficients' to the form which the Zonal Harmonic assumes when in place of cos 0, we write cos cos ' + sin o sin O'cos ($-$'). The name

“Kugelfunctionen” is employed by Heine, in his standard treatise on these functions, to designate Spherical Harmonics in general.

1. WE shall in this chapter regard a Zonal Solid Harmonic, of the degree i, as a homogeneous function of (oc* + y), and z, of the degree i, which satisfies the equation

d? V da V da V

+ = 0, dic + dy

dz Now, if this be transformed to polar co-ordinates, by writing rsin 8 cos % for x, r sin 8 sin for y, r cos @ for z, we

φ observe, in the first place, that a+y* = pa sin 0. Hence V will be independent of $, or will be a function of r and 6 only. The differential equation between r and o which it must therefore satisfy will be đ (TV)

dVi
+ sin e de sino =
drž
Ꮎ dᎾ

do
Now V, being a function of r of the degree i, may

be expressed in the form riP, where Pi is a function of 0 only. Hence this equation becomes

1 d

A sino de dᎾᏗ or, putting cos 0 = Ms

d

(

du In accordance with our definition of spherical surface harmonics, P, will be the zonal surface harmonic of the

(1),

{12 – way up to (i+1) P = 0 .......(2).

+

degree i. When it is necessary to particularise the variable involved in it, we shall write it P.().

The line from which is measured, or in other words for which u=1, is called the Axis of the system of Zonal Harmonics; and the point in which the positive direction of the axis meets a sphere whose centre is the origin of co-ordinates, and radius unity, is called the Pole of the system.

Any constant multiple of a zonal harmonic (solid or surface) is itself a zonal harmonic of the same order.

2. The zonal harmonic of the degree i, of which the line =1 is the axis, is a perfectly determinate function of Je, having nothing arbitrary but this constant. For the expression , may be expressed as a rational integral homogeneous function of r and 2, and therefore P, will be a rational integral function of cos 0, that is of M, of the degree i, and will involve none but positive integral powers

d

But P, is a particular integral of the equation

2. fcx)} x

+i (i+1) f (M) = 0......(3),

du and the most general form of f(u) must involve two arbitrary constants. Suppose then that the most general form of f(x) is represented by P, sody. We then have

)

d. f (d)
(1 - M) = (1 - 1) Jvdu + (1 – wa) Prv,

du
d.f(u)?

dP, (1 – u) du du

du ) – ?)

+

du

d

=

.

Hence, adding these two equations together, and observing that, since P, satisfies the equation (3), the coefficient

f sodu will be identically equal to 0, we obtain, for the de

of
termination of v, the equation

dv
P. ) du

=
do 2 dp, 2u
+

,
V (Po du T-M2
the integral of which is
log v + log P: (1 – pe) = log C, = a constant;

C,
P.; (1-) *

du
Hence

PY (1 -me")

– ; and we obtain, for the most general form of f(u),

du

.. vs

dimensions

, it may be seen that | 11 - .) P.

Now, Pr being a rational integral function of l of i

du

will assume the form of the sum of i+2 logarithms and i fractions, and therefore cannot be expressed as a rational integral function

du

are called Kugel

(1 - ?) P, functionen der zweiter Art by Heine, who has investigated their properties at great length. They have, as will hereafter be

seen, interesting applications to the attraction of a spheroid on an external point. We shall discuss their properties more fully hereafter.

3. We have thus shewn that the most general solution of equation (2) of the form of a rational integral function of u

of y. Expressions of the form Pila