5.T (2)

2 2

() +

(2) 6

+

+ ...).

.

)

12

2

16

8 cos 26 (1.2.3.4 &

7 (12)

4

....., ....),

(2)

3

(2)

87 cos 2¢' (1.7.3.4** *3.4.5.6m+

15

9.T 2) 13.Ta 2 cos 20'

+

1.2.3.4 3.4.5.6 5.6.7.8 Hence the potential of a spherical shell, of radius c and surface density cos 20', will be T.2) p2

gol4 (2) pe
+
+

+ c 3.4.5.6 c 5.6.7.8 c and

T c5 T (2) c?
+
+

+
p's

. 5.6.7.8 r? at an internal and external point respectively.

19. We will now explain the application of Spherical Harmonics to the determination of the potential of a homogeneous solid, nearly spherical in form.

The following investigation is taken from the Mécanique Céleste, Liv. III. Chap. II. Let r be the radius vector of such a solid, and let

pra

g=a +a (a, Y, +a, Y, + ... + a:Y, + ...), a being a small quantity, whose square and higher powers may be neglected, aq, Q,,...Az... lines of arbitrary length, and Y,, Y,,... Y... surface harmonics of the order 1, 2,....... respectively.

4
The volume of the solid will be ma'.

3
For it is equal to

? du

Again, if the centre of gravity of the solid be taken as origin, a, = 0.

For if z be the distance of the centre of gravity from the plane of xy, 4

Ti a z 3

a*+ 4a'a (a,Y,+u, Y, +...+c;Y; + ...) dudø

2

2

Now Y, is an expression of the form

Au + B (1 – vi?,cos $ + C(1–49)} sin 0, and therefore all the expressions , y, z cannot be equal to 0, unless

a

= 0. We may therefore, taking the centre of gravity as origin, write

prata

g=a +ala, Y,+ ... + a,Y,+ ...), the equation of the bounding surface of the solid.

Now this solid may be considered as made up of a homogeneous sphere, radius a, and of a shell, whose thickness is

a (a,Y,+ +a; Y; +...). The potential of this shell, at least at points whose least distance from it is considerable compared with its thickness, will be the same as that of a shell whose thickness is aa, and density

as

a

...

2

2

a

Po being the density of the solid. Therefore the potential, for any external point, distant R from the centre, will be a

a, Ya
+ ?

+ +
3R
5 R:

+
21+1 R'+1

4προ

a, Y, a

2

2

....).

*PR2 + 2HP. (a* – R") or 24p(a*-*)

.

3

(ei

'2
5

a, Y, R. 2i + 1 qitit

...)

a;Y: R

?, R2

+

...).

The potential at any internal point, distant R from the centre, will be made up of the two portions 4

R2 a?

3 for the homogeneous sphere,

Y, R? 4πραα?

+ +

a for the shell, and will therefore be equal to R

ra + Απραα

+

+ 3

5 as 21 +1 atti

2i 20. If the solid, instead of being homogeneous, be made up of strata of different densities, the strata being concentric, and similar to the bounding surface of the solid, we may

с deduce an expression for its potential as follows. Let er be the radius vector of any stratum, p its density, r having the same value as in the last Article, and p being a function of c only. Then, dc being the mean thickness of the stratum, that is the difference between the values of c for its inner and outer surfaces, the potential of the stratum at an external point will be 4προδο cʻc fa, Y, c + 4πρα

t... R

5

7 R8

zi+in+...).

+ .... ...... (1).

To obtain the potential of the whole solid at an external point we must integrate this expression with respect to c, between the limits 0 and a, remembering that p is a function of c.

Again, the potential of the stratum, above considered, at an internal point will be

c'8c (a, Y, R* a, Y, RR 4προδο + 4πρα

5 ca

+

a

To obtain the potential of the whole solid at an internal point we must integrate the expression (1) with respect to c between the limits 0 and R, and the expression (2) with respect to c between the limits R and a, remembering in both cases that p is a function of c, and add the results together.

CHAPTER V.

SPHERICAL HARMONICS OF THE SECOND KIND.

1. We have already seen (Chap. II. Art. 2) that the differential equation of which P, is one solution, being of the second order, admits of another solution, viz.

du CP

P: (1 - Me) Now if u between the limits of integration be equal to + 1, or to any roots of the equation P = 0 (all of which roots lie between 1 and -1), the expression under the integral sign becomes infinite between the limits of integration. We can therefore only assign an intelligible meaning to this integral, by supposing is to be always between 1 and 0, or between - 1 and We will adopt the former supposition, and if we then put C=-1,' the expression

will be always positive. We may therefore define the expression

du P

P;'(-1)' as the zonal harmonic of the second kind, which we shall denote by Vi, or Q: (u), when it is necessary to specify the variables of which it is a function.

It will be observed that, if u be greater than 1, P, is always positive. Hence, on the same supposition, Qis always positive.

du 1 We see

P' (1-7 (1.6

. P?W?-1

i.e.

M

a

that Q.=S

log

Mt 1
M-l'