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and the different Motions of the Earth's Axis.

LEMMA V.

To determine the momentum of rotation of a given spheroidAPrtO/>, revolving uniformly about its axis P p, with a givenangular celerity.

Let ENF be an ordinate to the generating ellipsis AVap,parallel to the axis of rotation P 'p: make AO (perpendicularto P 'p) = a ; OP (= Op) — b ; ON = x ; EN —y ; and letp denote the femi-periphery of the circle whose radius is unity.

Then it will be, as i : 2 p:\x-. 2px, the periphery of the Fig. ncircle generated by the point N. Therefore 2 px x 2 y will bethe measure of the surface generated by the ordinate EF, in therevolution of the ellipsis about its axis P 'p: which, drawn intothe square of the distance ON, gives 4 pyx^ for the momentumof rotation of all the particles in the said surface : so thatthe fluent of — 4 pyx^x will be the true measure of the forceto be determined.

Now, by the property of the ellipsis, we have x % = ^ x bb -yy ;and consequently xx — : whence, by substituting

A x Fyy — yf in the room of its equal — x 3 i, our fluxion is

transformed to -f- xFyf — yf: whose fluent, when y — b,

is found equal to Off.

Since

4 pa % b

3

COROLLARY.

is known to express the mass or content of the

spheroid, the momentum of rotation of any spheroid aboutits axis appears, therefore, to be just the fame as would arise,if yths of the whole mass was to revolve at the distance of thehighest point (A) from the axis of motion.

PROBLEM V.,

To determine the alteration of the pofition of the ter res rialequator , arifng from the adlion of the fun on the whole mass of theearth , during an infant of time.

Let