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the same element. The product of the volume density of any element of the shell, into the thickness of the shell in the neighbourhood of that element, is called "surface density.” We see from the above that, if the surface density be expressed by the series

0,P+0,P+o,P, + ... +0,Pi + ..., the potentials at an internal and an external point will seve

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10P,5

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1 0;P
+

+ 5

2i +1 This variation in surface density may be obtained either by combining a variable volume density with an uniform thickness, as we have supposed, or by combining a variable thickness with a uniform volume density, or by varying both thickness and density.

8. We have seen, in Chap. II., that any positive integral power of y, and therefore of course any rational integral

M, function of M, may be expressed by a finite series of zonal harmonics. It follows, therefore, that we can determine the potential of a spherical shell, whose density is any rational integral function of u.

Suppose, for instance, we have a shell whose density varies as the square of the distance from a diametral plane. Taking this plane as that of xy, the density may be expressed by pu, or pm. We have seen (Chap. II. Art. 20) that

1
pe = (1 +2P.).

3 Hence, by the result of the last Article, the potential will be

U/1 2 P, р

3 5 78

en ( + ") at an internal point,

U/1 2 P,64
pg C+50) at an external point;
р

3 r zou

3u - 1 322 -70% or, since Pogo?

22.

we obtain 2

2 U 1 1 322 5

for the potential at an internal point, 73

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Pg 16

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9. As an example of the case in which the density is represented by an infinite series of zonal harmonics, suppose we wish to investigate the potential of a spherical shell, whose density varies as the distance from a diameter. Taking this diameter as the axis of z, the density will be represented by p sin 0, or p (1 – ?)? We have investigated in Chap. 11.

р Art. 21, the expansion of sin in an infinite series of zonal harmonics. Employing this expansion, we shall obtain for the potential πρU1 1

1.3...(2-1) 1.3...(2-3) P

P 2 b 2 16 64 2.4...ii+2) * 2.4...(1–2)

or
(1 P. 1 72 1.3...(i-1) 1.3...(2-3

zi
Р

P. 2 2 r 1628

2.4...(2+2) ' 2.4...(2-2) i according as the attracted point is internal or external to the spherical shell, i being any even integer. All these expressions may be obtained in terms of surface density, by writing, instead of pU, 4mco.

10. We may next proceed to shew how the potential of a spherical shell of finite thickness, whose density is any solid zonal harmonic, may be determined. Suppose, for instance, that we have a shell of external radius a, and internal radius a', whose density, at the distance c from the centre, is his P.c", h being any line of constant length.

Dividing the sphere into concentric thin spherical shells, of thickness dc, the potential of any one of these shells, of

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goiti

а

radius c, at an internal point distant r from the centre will

pce be obtained by writing c for b, for C, 4tc dc for U, in

が the first result of Art. 6. This gives

р
4mc dc Picni 47

р

Pircdc. h 21 +1 +1 2i +1h To obtain the potential of the whole shell, we must integrate this expression, with respect to c, between the limits u' and a. This gives

or

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2i

4T P

or

c2i+2 piti

2i+3

(arita

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gith

2+1 Again, the potential of the shell of radius c, at an external

C point, will be

р
4c dc P.

P.

dc. 121 + 1 guit1 2i + 1 h

+1 Integrating as before, we obtain for the potential of the whole shell,

4T

р
P.;

a“) (2i + 1) (2i+ 3) h' Suppose now that we wish to obtain the potential of the whole shell at a point forming a part of its mass, distant r from the centre. We shall obtain this by considering separately the two shells into which it may be divided, the exterpal radius of the one, and the internal radius of the other, being each r. Writing r for a', in the first of the foregoing results, we obtain

21 pP,

' (a– po) 7P.

21 +1 h
And writing r for a in the other result, we obtain

47

(2i + 1) (2i + 3) h Adding these, we get for the potential of the whole sphere 47 PP: (a' – po

p2i+8 – a?

zt
2i + 1 hi 2 (2i + 3) pitis

1 2i+3 -a

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2i+3

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i+m+2

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2i+1 km P,

piti

It is hardly necessary to observe that the corresponding results for a solid sphere may be obtained from the foregoing, by putting a' = 0. If the density, instead of being P, c', be Prc",

р

р

Picm, similar hi

h" reasoning will give us, for the potential of the thin shell of radius c and thickness dc at an internal and external point respectively, 4T

477

+m

cit Pipoch-iti dc, and

dc. 2i + 1 h

" And, integrating as before, we obtain for the potential of the whole shell, 477 han Pe{a**** – a *m***8) x' at an internal point,

-s+2 (2i+ 1)(m - + 2) h 47 р

at an external point. (2i + 1)(m +2+3) h" And, at a point forming a part of the mass, 47 pP; la

-gom-it?

gom+i+3 m+i+31 2i +1h

-ľ + 2

m+i+ 3

р

m-it2

Pola"

mtit3 m+i+3

- a

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m

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m-i+2

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-a

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11.

2

Suppose, for example, that we wish to determine, in each of the three cases, the potential of a spherical shell whose external and internal radii are a, a', respectively, and whose density varies as the square of the distance from a diametral plane. Taking this plane as that of xy, the density may be ex

2P,+1 р

р pressed byz, z", or the cou". Now us

Hence the ha ha

3 density of this sphere may be expressed as

2 p lp

Pc 3

3 ha The several potentials due to the former term will be,

2 writing 2 for i and multiplying by

3'

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And for the latter term, writing 0 for i, and 2 for m, and

1 multiplying by

3'
4πρ

47. a' a's
р

- pa
р

pb
(a* a'),
12 h
15 ha

4

5r

15

4T P

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12. We may now prove that by means of an infinite series of zonal harmonics we may express any function of u what

M ever, even a discontinuous function. Suppose, for instance, that we wish to express a function which shall be equal to A from p=1 to j = 1, and to B from u=1 to u=-1. Consider what will be the potential of a spherical shell, radius C, of uniform thickness, whose density is equal to A for the part corresponding to values of u between 1 and X, and to B for the part corresponding to values of pe between a

λ and - 1.

Divide the shell, as before, into indefinitely narrow strips by parallel planes, the distance between any two successive planes being cd plo

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