Page images
PDF
EPUB

since this is the form which equation (2) assumes when V is an homogeneous function of the degree i

Now, put Vi+1 U,, and this becomes

=

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small]

Now, since U is a homogeneous function of the degree

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

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 V, by dividing by it, and conversely, that every form of V, can be obtained from U, by multiplying by 2+1 Such an expression as V, we shall call a Solid Spherical Harmonic of the degree . The result obtained by dividing by r, which will be a function of two independent variables and only, we shall call a Surface Spherical Harmonic of the same degree. A very important form of spherical harmonics 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 x2+ y2, or, will be functions of x2+ y and z. Harmonics independent of pare 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 @ sin e' cos (-'). The name "Kugelfunctionen" is employed by Heine, in his standard treatise on these functions, to designate Spherical Harmonics in general.

CHAPTER II.

ZONAL HARMONICS.

1. WE shall in this chapter regard a Zonal Solid Harmonic, of the degree i, as a homogeneous function of (x2 + y2)3, and z, of the degree ¿, which satisfies the equation d2V d3V d2 V + + dy

dx2

dz

0.

Now, if this be transformed to polar co-ordinates, by writing r sin cos o for x, r sin 0 sin & for y, r cos 0 for z, we observe, in the first place, that x+y=r2 sin20. Hence V will be independent of p, or will be a function of r and only. The differential equation between r and which it must therefore satisfy will be

[blocks in formation]

Now V, being a function of r of the degree i, may be expressed in the form rP, where P is a function of only. Hence this equation becomes

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

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

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

The line from which is measured, or in other words for which μ=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 μ1 is the axis, is a perfectly determinate function of μ, having nothing arbitrary but this constant. For the expression P may be expressed as a rational integral homogeneous function of r and z, and therefore P will be a rational integral function of cos 0, that is of μ, of the degree i, and will involve none but positive integral powers of μ.

But P is a particular integral of the equation

[merged small][merged small][ocr errors][merged small][merged small]

and the most general form of f(u) must involve two arbitrary constants. Suppose then that the most general form of ƒ (μ) is represented by Pvd. We then have

[merged small][subsumed][ocr errors][merged small][merged small][merged small][ocr errors][subsumed][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[ocr errors]
[ocr errors]
[ocr errors]

dP

αμ

+ 2 (1 − μ3) d2 v + P, d {(1 − 4 ) . }.
i
άμ du

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

of fudμ will be identically equal to 0, we obtain, for the de

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][ocr errors][ocr errors][subsumed][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

log v + log P2 (1 − μ2) = log C1 = a constant ;

Hence

[ocr errors][ocr errors][subsumed][merged small][merged small]

and we obtain, for the most general form of ƒ (μ),

[merged small][merged small][ocr errors]

Now, P. being a rational integral function of μ of i

[blocks in formation]

αμ
(1 − μ3) P2

will assume the

form of the sum of i+2 logarithms and i fractions, and therefore cannot be expressed as a rational integral function of μ. Expressions of the form P.(-) P αμ are called Kugel

2

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

« PreviousContinue »