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-w2)y, W(C-w2)z dynes; 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-pАx2-p(B-Q2)y2 — {p(C — N2)z2 ; so that at the surface the change of pressure is p(w2 — N2) (y2+22). If there is a liquid nucleus of density p+p, it can assume the form of the coaxial ellipsoid of semi-axes a, b, c, determined by the condition that is constant over its surface, the suffixes referring to this interior ellipsoid; and therefore απγραbc4' + πγρια 0,14') - 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 с w2 and if there is a central nucleus of density p+p1 bounded by the coaxial elliptic cylinder of semi-axes a, b1, the condition of equilibrium of the surface is or b22 c+b 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. This is the small calorie, also called the therm; as the calorie is sometimes defined as the quantity required to raise one kg of water through 1°C; this is 1000 therms. To be precise the water should be at or near its maximum density, or at a temperature of 4° C. 415. Different substances require different quantities of heat to raise their temperatures through the same number of degrees; and are thus distinguished by their specific heat. The specific heat (S.H.) of a substance is the number of B.T.U. required to raise the temperature of one lb of the substance through 1° F, or of calories required to heat one g through 1o C. In other words, the specific heat is the ratio of the quantity of heat required to heat the substance to the quantity required to heat an equal weight of water through the same number of degrees; the specific heat is thus the same in any system of units. With solid or liquid substances the specific heat is practically independent of the pressure or temperature, so that the above definition is sufficient for them; and now if weights W1, W2, ..., Wn (lb or g), of substances (solid or liquid) of S.H.'s C1, C2, ..., Cn, at temperatures T1, T2, ..., Tn (F or C) are placed in a vessel impervious to heat, the final uniform temperature T assumed by conduction is given by (W12+ W2c2+...+Wnen)T= W111+ W2C2T2 + ... + WnCnT T-Wer/WC. or = But substances in the gaseous state absorb or give out heat in a manner depending on the relation between the volume, pressure, and temperature, and the specific heat may be made to assume any value by a properly assigned relation, which must therefore be specified in defining the specific heat; for instance, the assigned relation may be of constant volume, or of constant pressure. 416. In melting a lb or g of a solid substance, although the temperature does not vary, a certain number of units of heat disappear, called the latent heat of fusion; and again, in converting the substance into vapour, the number of units of heat required is called the latent heat of vaporisation. The latent heat of fusion of ice into water is found to be 144 B.T.U. or 80 calories; and of vaporisation into steam at 212 F or 100 C is found to be about 966 B.T.U. or 537 calories. Suppose for instance that a meteor weighing 3 tons, of S.H. 0.2, heated to 3,000 F, fell into a pond containing 10 tons of water at 60 F; then a tons of water would be boiled away, given by 966x+10(212− 60) = 3 × 0·2 × (3000-212), x=0·158. If the water was at the freezing point, and one ton was frozen into ice, the temperature would be raised by the meteor to 210 F; and if the meteor weighed 4 tons, about 0-3 tons of water would be boiled away. According to Regnault's experiments, the latent heat of steam at any other temperature For C is 10917-0-695(F-32), B.T.U., or 6065-0695 C, calories; so that to heat one lb or g, respectively, of water from the freezing point, and to evaporate it into steam at temperature F or C requires 10917+0305(F-32), B.T.U., or 6065+0305 C, calories; this is called the total heat of steam at that temperature. |