276 SIPHONS ON A LARGE SCALE. In the siphons of fig. 61 the branches are vertical; and now if a, b denote the lengths of the vertical branches, and c the length of the horizontal part, then in the critical case (a+c)h=(b+c− a)(h—a), or b=(a+c)h +a-c. 194. As employed for drawing off water over an embankment, the siphon is shown in fig. 63; for example, over the reservoir dam of water works (fig. 20), or in draining a fen or inundation. (Proc. Inst. Civil Engineers, XXII.) An automatic valve, opening inwards, is placed at A and a stop valve at C. The siphon is filled either through a funnel by means of a hand pump, or else by exhausting the air by an air pump at B. On opening the stop valve C, the water flows through the siphon; and on closing the stop valve, the siphon remains filled for an indefinite time, the valve at A preventing the return of the water in AB. In this, as in all other cases, the height of B above the upper level of the liquid must be kept below the head of liquid corresponding to the atmospheric pressure. Sometimes the siphon is inverted, as required for carrying a water main across the bed of a river; and now there is no limitation of depth to its working. A water main, or a pipe line for conveying oil, carried in an undulating line in the ground, may be considered as a series of erect and inverted siphons; and on an emergency, the pipe may be carried over an obstacle, which is higher than the supply source or hydraulic gradient by something under the atmospheric head of the liquid. THE INTERMITTENT SIPHON. 277 195. An intermittent siphon is shown in fig. 64; the vessel is gradually filled up to the level of B, when the action of the siphon suddenly commences, and the vessel is rapidly emptied; and so the operation goes on periodically. The Cup of Tantalus, invented by Hero, depends on this principle; and it is also used for securing an intermittent scouring flow of water. The action of natural intermittent springs and geysers is explained in this manner; and the underground flow of certain rivers, such as the Mole, by subterranean inverted siphons. Examples. (1) If a vessel contains liquids of various densities, will the action of the siphon be impeded? Two equal cylindrical pails of horizontal section A are placed, one on the ground, and the other on a stand of height h; the former is empty, and the latter contains masses m, m2 of two different homogeneous liquids; a fine siphon tube of negligible volume has its two ends at the bottoms of the two pails and through it flows liquid until equilibrium is attained, a mass mg of density p remaining in the upper pail; prove that mï +m2−2m3 = Ahp. (2) A siphon tube with vertical arms filled with mercury, of S.G. σ, and closed at both ends is inserted into a basin of water. When the stoppers are removed, examine what will ensue, and prove the following results if the barometer is sufficiently high : (1) If b, the whole length of the outside arm, exceeds a, the whole length of the immersed arm, the mercury will flow outwards and the water will follow it. (2) If a > b, the end of the immersed tube must be at a depth below the free surface of the water exceeding (a−b)o in order that the mercury may not flow back into the basin. (3) Two equal cylinders side by side contain mercury, one quite full and open at the top, the other full to 20 inches from the top and closed, the 20 inches being occupied by air at the atmospheric pressure, which is 30 inches of the barometric column. If the two vessels are connected by a siphon dipping into the two liquids, prove that, when the siphon is put in action, 5 inches of mercury will flow from one of the cylinders into the other. What takes place when the leg of the siphon which is in the closed cylinder is not long enough to reach the mercury in that cylinder? CHAPTER VII. PNEUMATICS. THE GASEOUS LAWS. 196. Hitherto we have dealt with the properties of Liquids or Incompressible Fluids like Water; and now we proceed to consider Air and Gases, or Compressible Fluids, and their properties, a branch of Hydrostatics sometimes called Pneumatics, from the Greek word πveυματική, meaning the science which concerns πνεῦμα, air or gas. A given quantity of a Gas ("a parcel of gas" in Boyle's words) requires to be kept in a closed vessel, to prevent diffusion; and by changing the volume of the vessel and the temperature, the pressure of the gas is altered. Given the volume and the temperature, the pressure of a given quantity of a gas is determinate; so that the pressure p is a function of the volume v or density p, and of the temperature 7. or Expressed analytically p=f(v, 7), F(p, v, 7) = 0 ; and to determine this function, two new Laws, based upon experiment, are required, which are called 280 THE GASEOUS LAWS 197. The Gaseous Laws. LAW I.-BOYLE'S LAW. "At constant temperature the pressure of a given quantity of a Gas is inversely proportional to the volume, or directly to the density." This law was enunciated by Boyle in his Defence of the Doctrine touching the Spring and Weight of the Air in answer to Linus, 1662; abroad it is attributed to Mariotte, who did not however publish it till 1676. Thus if p denotes the pressure and v the volume of unit quantity of the gas, one gramme suppose, and p denotes the density, so that p=1/v, then p=kp, or pv=k, where depends only on the temperature: so that, on the (p, v) diagram, an isothermal is a hyperbola (fig. 65), along which the hydrostatic energy pv (§ 14) is constant. For instance, a guuner, who can push with a force of P pounds, can, under an atmospheric pressure of p lb/in2, introduce an airtight sponge into a closed cannon, d ins in calibre and ins long in the bore, a distance x ins, given by Thus, if P=100, p=15, d=5, l=120, we find that x=30.42. LAW II.-CHARLES'S OR GAY-LUSSAC'S LAW. "At constant pressure the volume of the Gas increases uniformly with the temperature, and at the same rate for all gases." Combining this with Boyle's Law we find that the product of the pressure and volume of a given quantity of |