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

a2-b2 = a2 -ẞ2, a2 - c2 = a2 - y2............ (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. is Ivory's Theorem.

This

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

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

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

— if {(ƒ' — x)2 + (g' − y)2 + (h' − 2)'}] dxdy

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

=

= pabsƒ [† {(ƒ' — ar)2 + (g' − bs)2 + (h' — ct)2}

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

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

Now (f'ar)2 + (g' − bs)2 + (h' ± ct)2

12

12

=ƒ"2+g"2 + h'2 − 2 (f'ar + g'bs + h'ct) + a2r2 + b2s2 + c2t2,

substituting for h" by (3) and for ť2,

=ƒ" (1-2)+9" (1 − 2) + v2 - 2 (f'ar+g'bs ± h'ct)

+(a2 − c2) p2 + (b2 — c2) s2 + c3,

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

(a2 — c3) + 22 (b2 — c2) + c2 − 2 ( far +gßs ± hyt)

= ƒ2 + g2 + h2 − 2 (für +gßs ± hyt) + a2r2 + ß2s2 + y2ť2, by (3),

= (ƒ — 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}

− ¥ {(ƒ − ar)2 + (g − Bs)2 + (h − yt)"}] drds

ab

ав

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

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 equa

tions are

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

an equation which can be satisfied only by x = 0, y = 0, z=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

CHAPTER II.

LAPLACE'S COEFFICIENTS AND FUNCTIONS.

17. IN the present Chapter we shall develop the properties of those remarkable quantities which have received the name of their great discoverer, under the designation of LAPLACE'S COEFFICIENTS AND FUNCTIONS. To do this it will be necessary to anticipate the subject of the following Chapter, and to bring in here a Proposition which should properly stand at the head of that division of this treatise.

PROP. To obtain formulæ for the calculation of the attraction of a heterogeneous mass upon any particle.

18. Let p be the density of the body at the point (xyz); fgh the co-ordinates of the attracted particle; and, as before, suppose that A, B, C are the attractions parallel to the axes x, y, z. Then

the limits being determined by the equation to the surface of

19. It follows, then, that the calculation of the attractions A, B, C depends upon that of V. This function cannot be

calculated except when expanded into a series. It is a function of great importance in Physics: and, for the sake of a name, has been denominated the Potential of the attracting mass, as upon its value the amount of the attractive force of the body depends.

20. As the axes and origin of co-ordinates in the previous Article are altogether arbitrary, it follows that if r be the distance of the attracted point from any fixed point in the attracting body, then the attraction in the line of r, towards the origin of r,

=

dV
dr

d2 V
df2 dg2 dh2

cording as the attracted particle is not or is part of the mass itself; p being the density of the attracted particle in the

latter case.

21. By differentiating V, we have

dv

df

d2 V

́p {2 (ƒ — x)2 — (g − y)2 — (h — z)2} dx dy dz
{(ƒ − x)2 + (g− y)2 + (h — z)2} §

a = = /// f = — −

In the same manner we shall have

d2 V

dg

'p {2 (g − y)2 — (ƒ— x)2 — (h — 2)2} dx dydz
{(f−x)2+(gy)2 + (h — 2)2 } §

dr2 = [[[ P ( — —

2

2 (h — z)2 — ( ƒ — x)2 — (g — y)2} dx dy dz
{(ƒ −x)2 + (g− y)2 + (h − 2)2} §

When the attracted particle is not a portion of the attracting mass itself, then xyz will never equal fgh respectively,