Let-

and the different Motions of the Earth's Axis. 3

r a — scmi-diam. OA' (perpend, to the plane PA Ope), Fig. l.j A— area of the ellipse PAO pa,

| y = force acting on a particle at the remotest point A',

I x = ON, the dish of any section DQNEQfrom VAOfa:

Then, this section being also an ellipse, similar to P Apa, weshall have, by the property of the ellipsis, as A 'Q 1 (aa) : A'O 1— — xx) : : PO 1 : DN* : : the area PAO pa ( A ) to

the area DQNEQj== A x ——— (by the property of similar

figures). Hence it is evident that A x

act — xx

x — x y wilt be

the sum of all the forces whereby the particles in the ellipseDQEQ^_are urged parallel to the axis Vp of the spheroid;which quantity, drawn into (x) the length of the lever ON,will, consequently, express the effect of all the said forces toturn the spheroid about its center : and so the fluent of

A x —-— x — x xx x y, which is A x — x y (when x = a)

will truly express one half of the quantity sought.

COROLLARY.

If the mass, or content of the spheroid, which is A x 2ax —,

be denoted by S ; then the force A x — x y, whereby the-

spheroid tends to turn about its center, will be truly definedby jS x a x y, which therefore is just }th part of what itwould be, if all the particles were to act at the distance ofthe remotest point A'.

LEMMA II.

Suppose a body to revolve in the circumference of a circle AEsE, Fig. 2 .•whilst the circle itself turns uniformly about one of its diametersAa, as an axis , with a very sow motion ; it is proposed to deter-mine the law of the force, aiding on the body in a direction per-pendicular to the plane of the circle, necessary to the continuation ofa motion thus compounded.

B 2

Let