* If the spheroid be prolate, c is a and the denominator of this must be written c2 (c2 - a2) cos2 0, and the integral would involve logarithms instead of circular arcs.

13. COR. 1. We gather from these expressions, that the attraction is independent of the magnitude of the spheroid, and depends solely upon its ellipticity. Hence the attraction of the spheroid similar to the given one, and passing through the attracted particle, is the same as that of any other similar concentric spheroid comprising the attracted particle in its mass. Hence a spheroidal shell, the surfaces of which are similar and concentric, attracts a point within it equally in all directions.

14. COR. 2. If we put the ellipticity of the spheroide, and suppose e so small that we may neglect its square, we have

If we had taken an ellipsoid instead of a spheroid, the expressions would not have been capable of integration.

15. If we had attempted to find the attraction on an external particle according to the process of the last Article, we should have fallen upon expressions which no known methods have yet integrated and therefore we are unable by any direct means to obtain the attraction of a spheroid on an external particle. Mr Ivory has, however, devised an indirect method of obtaining it, which we shall now proceed to develop. He has discovered a theorem by which the attraction of an ellipsoid upon an external particle is shown to be proportional to that of another ellipsoid, dependent on the first for form and dimensions, upon a particle internal to it, and therefore (in the case of a spheroid, or ellipsoid of revolution) determinable by the last Proposition.

PROP. To enunciate and prove Ivory's Theorem.

SPHEROID ON EXTERNAL POINT.

IVORY'S THEOREM. 11

be the equations to the surfaces of two ellipsoids having the same centre and foci: then

a2 - b2 = a2 - ẞ2, a2 - c2 = a2 - y3........................... (1).

Let fgh, f'g'h' be the co-ordinates to two particles so situated on the surfaces of these ellipsoids that

Also since (fgh) (f'g'h') are points in the surfaces of the first and second ellipsoids respectively, we have

Then the attraction of the first ellipsoid parallel to the axis of x on the particle at the point (f'g'h') on the surface of the second, is to the attraction of the second ellipsoid on the particle at the point (fgh) on the surface of the first in the same direction, as ab aß, the law of attraction being any function of the distance: and similarly with respect to the axes of y and z. This is Ivory's Theorem.

We shall, for convenience, represent the law of attraction by the function rẻ (r2), r being the distance.

The attraction of the first ellipsoid on the particle (ƒ'g'h') parallel to the axis of z

= p SSS(h' — z) $ {(ƒ' − x)2 + (g' − y)2 + (h' — z)2} dx dy dz,

√(1-22-21),

√(1-2),

-

α

a and a

and b

α

= pSS[↓ {(ƒ' − x)2 + (g' − y)2 + (h' + z)o}

— ¥ {(ƒ' — x)2 + (g' − y)2 + (h' — z)3}] dx dy

but we do not substitute this value merely that the function be preserved under as simple a form as possible. Now put x = ar, y=bs, z = ct, then the attraction

may

= pabs] [† {(ƒ' — ar)2 + (g' — bs)2 + (h' +ct)*}

− & {(ƒ' — ar)2 + (g' — bs)2 + (h' — ct)2}] dr ds,

the limits of s being — √(1 −2) and √(1 − r2), and those of r being 1 and 1: also t=√(1 − p2 — s2).

Now (f' — ar)2 + (g' — bs)2 + (h' ±ct)2

= ƒ"2 +g"2 + h22 − 2 (f'ar + g'bs ±h'ct) + a2r2 +b2s2 + c2t2, substituting for h' by (3) and for t,

12

= ƒa (1 − 2) + 9′′ (1 − Z ) + y2 − 2 (far+g'be± h'ct)

eliminating f'g'h' by (2) and making use of (1),

= f2 + g2 + h2 − 2 (fur + gßs ± hyt) +a2r2 + B2s2 + y2ť2, by (3),

=

· (f— ar)2 + (g − Bs)2 + (h ± yt).

Hence the attraction of the First Ellipsoid on (f'g'h') parallel to z,

=pab SS [↓ {(ƒ— ar)2 + (g − Bs)2 + (h+yt)2}

2

-&{(f―ar)2 + (g- Bs)2 + (h-yt)"}] dr ds

=

SPHEROID ON EXTERNAL POINT. IVORY'S THEOREM. 13

ab

x attraction of Second Ellipsoid on (fgh) in the same direction.

aß

The same may be proved for the attractions parallel to the other axes: and consequently the Theorem, as enunciated,

is true.

We may observe that one of these ellipsoids must necessarily be wholly within the other. For if not, the points in which they cut each other lie in the line of which the equations are

Suppose a less than a; the points of intersection must satisfy the equation

2=

an equation which can be satisfied only by x = 0, y =0, 0. But these do not satisfy the equations above; and therefore the surfaces do not intersect in any point.

To find the attraction of any ellipsoid of which the semiaxes are a, b, c upon an external point (f'g'h') by the help of this Theorem, we must first calculate the attraction of an ellipsoid of which the semi-axes are aßy, determined by equations (1) and the second of (3), on an internal point (fgh), f, g and h being given by equations (2). And then the attractions required will be those multiplied by