MACLAURIN'S AND JACOBI'S FIGURES.

501

408. One solution is obviously b2=c2 (Maclaurin's spheroid); and then

αλ

0

(a2+λ)3(c2+λ) ̄ ̄√(c2 — a2)

2

=

409. The ellipticity (§ 358) of the spheroid is given by

Thus if we assume that the Earth is homogeneous, and of mean radius R, then

agreeing very nearly with Newton's estimate of 230. (Principia, lib. iii., prop. xix.)

But as the true ellipticity of the Earth is about 1/300, the density must be greater in the interior; for if we assume that there is a concentric centrobaric nucleus of

502

EQUILIBRIUM OF ROTATING LIQUID.

density up and radius k, the additional potential at

so that equation (A) of surface equilibrium becomes

{πyp(u−1)1⁄2 ~ — } Aa2={πyp(μ−1)22
πγρ(μ-1) - πγρ(μ-1) -- {} C c2 + } w2c

a

g=πγρα+ πγρ(μ-1) R2'

and now

1

so that

=

4g

1+3(μ−1)(k/R)3

€ 5R2 1+ (-1) (k/R) 3'

410. The integral which is the remaining factor of § 407, gives the relation connecting b/a and c/a f Jacobian ellipsoid.

Putting bc in this integral gives

ΟΙ

(a2 +λ)1(c2+λ)3

a transcendental relation which becomes

(3+14f2+3f4)tan-If=3f+13ƒ3,

At this critical angular velocity the stable fig of equilibrium of the rotating liquid will pass Maclaurin's spheroids into Jacobi's ellipsoids (Tho and Tait, Natural Philosophy, $$ 771-778).

JACOBI'S ROTATING ELLIPSOID.

503

411. A plummet, weighing W g, at the end of a plumb line on the surface of Jacobi's ellipsoid, will experience an apparent attraction of gravitation, having components WAx, W(B-2)y, W(C-w2)2 dynes;

and these may be written

WAa2(px py pz2\

Ρ a2, b2, c2)'

where p denotes the length of the perpendicular from the centre on the tangent plane; so that the plumb line will take the direction of the normal to the ellipsoid; and denoting the polar gravity by G, and the length of the normal to the equatorial plane by v, the tension in dynes of the plumb line,

Wg= WAa2/p= WAv=WGv/a.

An ocean of small depth would spread itself over this ellipsoid, so that the depth at any point is inversely as g, and therefore directly as p.

412. If this Jacobian ellipsoid is enclosed in a rigid case, and rotated with new angular velocity N, then p=constant-pAx2-\p(В—N2)y2 — {p(C — N2)z2 ;

so that at the surface the change of pressure is \p(w2 — N2)(y2+z2).

If there is a liquid nucleus of density p+p, it can assume the form of the coaxial ellipsoid of semi-axes a1, b1, c1, determined by the condition that

Typ abc (P-A'x2 - B'y2 — 10′22)

πγρ

is constant over its surface, the suffixes referring to this interior ellipsoid; and therefore

504

ROTATING CYLINDERS OF LIQUID.

equations for determining A, B, C, etc., when A', B, C' and 2 are given.

Thus if the outer case is spherical,

A'B'C', and abcA'.

It might even be possible for the interior nucleus to rotate bodily as a concentric but not coaxial ellipsoid, when the outer case is made to rotate about an axis not a principal axis.

=

413. When a∞, the ellipsoidal case becomes an elliptic cylinder; and now

so that if filled with one liquid rotating bodily, the surfaces of equal pressure are the quadric cylinders given by

and if there is a central nucleus of density ptpi bounded by the coaxial elliptic cylinder of semi-axes a, b, the condition of equilibrium of the surface is

CHAPTER XIII.

THE MECHANICAL THEORY OF HEAT.

414. When work is done by the expansion of a gas, as, for instance, by the powder gases in the bore of a gun, or by the steam in the cylinder of a steam engine, a certain amount of heat is found to disappear; and according to the First Law of Thermodynamics, the heat which disappears bears a constant ratio to the work done by the expansion.

Thermodynamics is the science which investigates the relations between the quantities of heat expended and work given out in the Conversion of Heat into Work, and vice versa; and for a complete exposition of the subject, the reader is referred to the treatises of Clausius, Tait, Verdet, Maxwell, Shann, Baynes, Parker, Alexander, Anderson, etc.; also to the Smithsonian Index to the Literature of Thermodynamics.

In measuring quantities of heat, the unit employed is either the British Thermal Unit (B.T.U.) or the calorie. The B.T.U. is the quantity of heat required to raise the temperature of one lb of water through 1° F.

The calorie is the quantity of heat required to raise the temperature of one g of water through 1° C.