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V= 4mada P, (u) 9:(v) Pi

(a– C) 23. We may apply this result to the discussion of the following problem.

If the potential of a shell in the form of an ellipsoid of revolution about the greatest point be inversely proportional to the distance from one focus, find the potential at any internal point, and the density.

If the potential at P be inversely proportional to the distance from one focus S, and I be the other focus, we have,

HP+ SP=2n, HP-SP=2w,

... SP=n-w. Hence if M be the mass of the shell, V, the potential at any external point,

M

=

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Now, by what has just been seen, the internal potential, corresponding to Pi (a) Qi (v), is

a

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a

Qi

lla- 6,7
Pi(u) P: (v)

la – 62)
Hence, if V, be the potential at any internal point,

P

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a

Q
M

(a - b)
V,
£ (2i + 1)

P.(m) P. (v).
(a' - 636

P.

l(a621) And the volume density corresponding to Po(u) Q: (v) is

P (u)

a

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a

P.

(a* – 69) Hence the density corresponding to the present distribution is M

Pilu) p=

E (2i+1). 47 (a62,5 686

la’ 63)) If V, had varied inversely as HP, we should have had

M

n ta' and our results would have been obtained from the foregoing by changing the sign of w, and therefore of M.

Now, by adding these results together, we obtain the distributions of density, and internal potential, corresponding to M M


V.
+

M
η +ω

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S

24.

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= M multiplied by the axis of revolution of the confocal ellipsoid, and divided by the square on the conjugate semidiameter. We may express this by saying that the potential at any point on the ellipsoid is inversely proportional to the

a

Qui

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a

P,

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square on the conjugate semi-diameter, or directly as the square on the perpendicular on the tangent plane.

Corresponding to this, we shall have, writing 2k for i, since only even values of i will be retained,

) 2M

(a* - 62) 8 (4k+1)

P4 (u) P x (v), (a’ – 62)

lla- 6%)

6,4) 2M

Pm (u) pa

(4k + 1) 47 (a62)_686

PO

la – 62)})

? k being 0, or any positive integer. Again, subtracting these results we get M M

20 V

M η

na-w2" =M multiplied by the distance from the equatoreal plane, and divided by the square on the conjugate semi-diameter.

This gives, writing 2k +1 for i,

a

27

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2x+1

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Pati

Р

21

2M
E (4k+3)

PU(u)
p=
47 (a' 6)$686

la’ – 6°)?)

( 25. In attempting to discuss the problem analogous to this for an ellipsoid of revolution about its least axis, we see that since its foci are imaginary, the first problem would represent no real distribution. But if we suppose the external potential to be the sum or difference of two expressions, each inversely proportional to the distance from one focus, we

obtain a real distribution of potential—in the first case inversely proportional to the square on the conjugate semi-diameter, in the latter varying as the quotient of the distance from the equatoreal plane by the square on the conjugate semi-diameter.

It will be found, by a process exactly similar to that just adopted, that the distributions of internal potential, and density, respectively corresponding to these will be:

In the first case

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2M

922
V
E (4k+1)

P.A () P2 (v),
(a’ – c)

Pak

(a ) 2M

Px(u) p=

Σ (4k + 1) 47 (a’ – 039% ada

PA

(a– c)? k being 0, or any positive integer.

In the second case

2

1

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2x+1

c

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ex+1
2M
V = E (4k+3)

(a –c)

P. (a? -*)

23+7 () Pattz (v), P2x+1

(a? — c)}} 2M

() р

Σ (4k +3) 47 (a’ – cada

Pak+1

) k being 0, or any positive integer,

26. We may now resume the consideration of the ellipsoid with three unequal axes, and may shew how, when the potential at every point of the surface of an ellipsoidal shell is known, the functions which we are considering may be employed to determine its value at any internal or external point. We will begin by considering some special cases,

10

F. H.

by which the general principles of the method may be made more intelligible.

27. First, suppose that the potential at every point of the surface of the ellipsoid is proportional to x = suppose.

VC

a

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=

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In this case, since when substituted for V, satisfies the equation v?V= 0, we see that V will also be the potential at any internal point. But this value will not be admissible at external points, since x becomes infinite at an infinite distance. Now, transforming to elliptic co-ordinates

(€ + a) (v + a?) (v + a?)

(a* b) (a' -c) And the expression Sle+a”) (u+a”) (u'+ao)

df al (a?) (a? —co) (x+a){(x +a%)(4+62)(+co)}}

dy
(4+a”) {[%+a”) (24 +6) (+co)}}
satisfies, as has already been seen, the equation y?V=0, is
equal to V. at the surface of the ellipsoid, and vanishes
at an infinite distance. This is therefore the value of the
potential at any external point. It may of course be written
VEC

dy
(36 + aʼ) {(34 +a) (+b) (x+co)}}

df

(4 + a?) {(4 + a*) (+) (24 +co)}}' 28. Next, suppose that the potential at every point of the surface is proportional to yz=V,

yz

In this bc

suppose.

0

a

ox

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a

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