31. Any rational integral function V of x, y, 2, which satisfies the equation v' V = 0, can be expressed in a series of Ellipsoidal Harmonics of the degrees 0, 1, 2...i in x, y, z. For if V be of the degree i, the number of terms in V will

(i+1) (i+2) (+3) be

Now the condition V=0 is 6 equivalent to the condition that a certain function of x, y, z of the degree 2 – 2, vanishes identically, and this gives rise

(i-1) i (i +1) to

conditions. Hence the number of inde6 pendent constants in V is (i+1) (i + 2)(2+3) (i – 1) i (i+1) 6

6

2

or (i+1). And the number of ellipsoidal harmonics of the

1 3 degrees 0, 1, 2...i in x, y, z or of the degrees 0, 1,

2""'2

22 in e, v, u', is, as shewn in Arts. 6 to 10 of this Chapter,

1+3+5+...+ 2i +1, or (i + 1). Hence all the necessary conditions can be satisfied.

1

32. Again, suppose that attracting matter is distributed over the surface of an ellipsoidal shell according to a law of density expressed by any rational integral function of the co-ordinates. Let the dimensions of the highest term in this expression be i, then by multiplying every term, except those of the dimensions i and i - 1 by a suitable power of

2c2
a " " ?

we shall

express the density by the sum of two rational integral functions of x, y, z of the degrees i, i – 1, respectively. The number of terms in these will be (i+1) (i+2), (i+1)

i +

or (i + 1)? 2

2

+

And any ellipsoidal surface harmonic of the degree i, i – 2... in x, y, 2, may, by suitably introducing the factor

22322

a T 62 c' be expressed as a homogeneous function of x, y, z of the degree i ; also any such harmonics of the degree 2-1, 2-3... in x, y, z may be similarly expressed as a homogeneous function of x, y, z of the degree i – 1. And the total number of these expressions will, as just shewn, be (i+1)?, hence by assigning to them suitable coefficients, any distribution of density according to a rational integral function of x, y, z may be expressed by a series of surface ellipsoidal harmonics, and the potential at any internal or external point by the corresponding series of solid ellipsoidal harmonics.

33. Since any function of the co-ordinates of a point on the surface of a sphere may be expressed by means of a series of surface spherical harmonics, we may anticipate that any function of the elliptic co-ordinates v, u' may be expressed by a series of surface ellipsoidal harmonics. No general proof, however, appears yet to have been given of this proposition. But, assuming such a development to be possible at all, it may be shewn, by the aid of the proposition proved in Art. 15 of this Chapter, that it is possible in only one way, in exactly the same way as the corresponding proposition for a spherical surface is proved in Chap. IV. Art. 11.

The development may then be effected as follows. Denoting the several surface harmonics of the degree i in x, y, z,

in 2

u, by the symbols V. (1), V/(2),... V,(2+1), and by

v,
F(u, v) the expression to be developed, assume
F(u, v)= 0.V.+ C(1) V.(1) + C(2) V (2) + C (3) V7(8) +...

)
+C.MV(1) + ... + COV6 + ...
1) V7

)
Then multiplying by e Vco) and integrating all over the
surface, we have

or

1

1

1

seF (u, v) V[") dS = c«»se(Vm) e as

The values of sef (u, v)V_(a) ds, and of se(v)*ds must

)

be ascertained by introducing the rectangular co-ordinates 2, 4, 2, or in any other way which may be suitable for the particular case. The coefficients denoted by C are thus determined, and the development effected.

EXAMPLES.

+

9

3

10

n-1)

n

8 16 8 1. Prove that (sin )

15

21 35 Why cannot (sin 6) be expanded in a finite series of spherical harmonics ?

0

1 + sin
1
2. Prove that 1 + +

P.+ PA +... = log
3

sin

2 3. Establish the equations

dP u 1)

du

NP=(2n-1) dd -, -(~– 1)P,-1 4. If u = cos 0, prove that

A

2 + m ^3 + + ... + (-1)

2 and also that

A P.(u)=(-1)+(- 1)*+1 i(i + 1) cos®

2

li+m +(-1)"+

a m) i 5. Prove that, if a be greater than c, and i any odd integer greater than m,

1 2 du =

Σ

m

P. (-) = 1 –i (i+1) sin

(sin 9+ .

)('++1) +...

2 + m

m+1cm 2m

6.

i(i-1)

2

.

(u – 1)

9. Prove the following equation, giving any Laplace's coefficient in terms of the preceding one:

where Cp=pepe' + 1 - * V1 – pe” cos (w-w) and C is zero if ñ be

even, and

n+1

2

n1

In + 1 (-1)

if n be odd.

2"+l{1} (n + 1)}?! 10. If i, j, k be three positive integers whose sum is even,

prove that

| P.PP/

+

1.3... (j+k-i-1) 1.3... (k+i-j-1) 1.3... (i+j-k – 1) 2.4 ... (j+k-1) jki 2.4 ... (k+i-j) 2.4... (i+j-k)

j 2.4...(i+j+k)

1

1.3... (i + j + k – 1) i+j+k+1 Hence deduce the expansion of P.P, in a series of zonal harmonics. 11. Express a'y + y + y2 + y +z as a sum of spherical

% harmonics.

12. Find all the independent symmetrical complete harmonics of the third degree and of the fifth negative degree.

13. Matter is distributed in an indefinitely thin stratum over the surface of a sphere whose radius is unity, in such a manner that the quantity of matter laid on an element (SS) of the surface is

8S (1 + ax + by + c2 + fo* + gy® + hzo),