h = 10000 (1 + ·00244 (t + t' 2 b ß × (1 + n (q—q')' If the centigrade thermometer is used, because the beginning of the scale agrees with the temperature f, so that f= 0, the formula is more simple; and if the expansion for air and mercury be both adapted to the degrees of this scale, h = 10000 (1 + 00441 (* + *') log. 2 b B(100018 (q — q')' 6. The temperature of the air diminishes on ascending into the atmosphere, both on account of the greater distance from the earth, the principal source of its heat, and the greater power of absorbing heat that air acquires, by being less compressed. 7. Professor Leslie, in the notes on his Elements of Geometry, p. 495, edit. 2d, has given a formula for determining the temperature of any stratum of air when the height of the mercury in the barometer is given. The column of mercury at the lower of two stations being b, and at the upper 3, the diminu tion of heat, in degrees of the centigrade, is This seems to agree well with observation. b 3 25. 8. If the atmosphere were reduced to a body of the same density which it has at the surface of the earth, and of the same temperature, the height to which it would extend is, in fathoms, Hence if b be the height of the mercury in the barometer, reduced to the temperature t, the specific gravity of mercury The divisor 72 is introduced in consequence of b being expressed in inches.-(Playfair's Outlines.) SECTION II.-Pumps. 1. Def. The term Pump is generally applied to a machine for raising water by means of the air's pressure. D 2. The common suction-pump consists of two hollow cylinders, which have the same axis, and are joined in A C. The lower is partly immersed, perpendicularly in a spring or reservoir, and is called the suction-tube; the upper the body of the pump. At a c is a fixed sucker containing a valve which opens upwards, and is less than 34 feet from the surface of the water. In the body of the pump is a piston D made air-tight, moveable by a rod and handle, and containing a valve opening upwards. And a spout & is placed at a distance greater or less, as convenience may require, above the greatest elevation of D. 3. To explain the action of this pump. B C Suppose the moveable piston D at its lowest depression, the cylinders free from water, and the air in its natural state. On raising this piston, the pressure of the air above it keeping its valve closed, the air in the lower cylinder A в forces open the valve at A c, and occupies a larger space, viz., between в the surface of the water, and D; its elastic force therefore being diminished, and no longer able to sustain the pressure of the external air, this latter forces up a portion of the water into the cylinder A B to restore the equilibrium. This continues till the piston has reached its greatest elevation, when the valve at A C closes. In its subsequent descent, the air below D becoming condensed, keeps the valve at A c closed, and escapes by forcing open that at D, till the piston has reached its greatest depression. In the following turns a similar effect is produced, till at length the water rising in the cylinder forces open the valve at A c, and enters the body of the pump; when, by the descent of D, the valve in a c is kept closed, and the water rises through that in D, which on re-ascending, carries it forward, and throws it out at the spout G. 4. Cor. 1. The greatest height to which the water can be raised in the common pump by a single sucker is when the column is in equilibrio with the weight of the atmosphere, that is, between 32 and 36 feet. 5. Cor. 2. The quantity of water discharged in a given time is determined by considering that at each stroke of the piston a quantity is discharged equal to a cylinder whose base is a section of the pump, and altitude the play of the piston. = = 6. To determine the force necessary to overcome the resistance experienced by the piston in ascending. Let h the height HF of the surface of the water in the body of the pump above EF the level of the reservoir; and a2 the area of the section м N. Let h' the height of the column of water equivalent to the pressure of the atmosphere; and suppose the piston in ascending to arrive at any position m n which corresponds to the height I F. It is evident that the piston is acted upon downwards by the pressure of the atmosphere = a h', and by the pressure of the column в m = a2 × HI; therefore the whole tendency of the piston to descend = a3 (h' + H I.) But the piston is acted upon upwards by the pressure of the air on the external surface EF of the reservoir a h'; part of which is destroyed by the weight of the column of water having for its base m n, and height F 1; the whole action upwards a2 × (h' F 1); whence F = a2. (h' + 1) = a2. (h' — F 1) = = a*. F H = a2 h, that is, the piston throughout its ascent is opposed by a force equal to the weight of a column of water having the same base as the piston, and an altitude equal to that of the surface of the water in the body of the pump above that in the reservoir. In order, therefore, to produce the upward motion of the piston, a force must be employed equal to that determined above, together with the weight of the piston and rod, and the resistance which the piston may experience in consequence of the friction against the inner surface of the tube.* When the piston begins to descend, it will descend by its own weight; the only resistance it meets with being friction, and a slight impact against the water. 7. Cor. 1 If the water has not reached the piston, let its Suppose the body of the pump to be 6 inches in diameter, and the greatest height to which the water is raised to be 30 feet; suppose, also, the weight of the piston and its rod to be 10lbs., and the friction one-fifth of the whole weight. Then, by the rule at p. 201, 1 of the square of the diameter gives the ale gallons in a yard in length of the cylinder, and an ale gallon, p. 290, weighs 10 lbs. Therefore (62 × 10) + 1 62 × 10)= 360+7·4 = 367-4 lbs. weight of the opposing column of water. And 3674+10+ (377-4)=452.9 lbs., whole opposing pres sure. 30 If the piston rod be moved by a lever whose arms are as 10 to 1, this pressure will be balanced by a force of 45-29 lbs., and overcome by any greater force. level be in v z. The under surface of the piston will be pressed by the internal rarefied air. But this air, together with the column of water, E v, is in equilibrio with the pressure of the atmosphere a h'; and.. its pressure a2. (h' — E v). And the pressure downwards a h'; . Hence the force requisite to keep the piston in equilibrio increases as the water rises and becomes constant, and = a2h as soon as the water reaches the constant level B н. 8. Cor. 2. If the weight of the piston be taken into the account, let this weight be equal to that of a colump of water. whose base is m n and height p, =a2 p; .. F = a2. (Ev+p). 9. To determine the height to which the water will rise after one motion of the piston; the fixed sucker being placed at the junction of the suction-tube and body of the pump supposing that after every elevation of the piston there is an equilibrium between the pressure of the atmosphere on the surface of the water in the reservoir, and the elastic force of the rarefied air between the piston and surface of the column of water in the tube, together with the weight of that. column. Let a b be the surface of the water in the sucfion-tube, after the first stroke of the piston if the piston were for an instant stationary at D, the pressure of the atmosphere would balance E b, and the elastic force of the air in N a. Let A E the height of the suction-tube = a, DR the play of the piston = b, D R a EN 1 1 a E F h the height of a column of water equivalent to the pressure of the atmosphere, y = the height of a column equivalent to the pressure of the air in N α, x = E α, and R and r = the radii of the body and the suctiontube. .. h R2 b+hr2 a — h r2 x=R2 b x+r2 a x — r2x2+h r2 a, R3 R3 b 2.2 b, and y = }. {2 h—p √p2-4 h m b}, only one of which values will be applicable, viz. that which answers to the lower sign; since x and y must be less than h; and if the upper sign be used, a will be found greater than h. 10. Having given the height of the water raised, and that due to the pressure of air in the pump after the first ascent of the piston; to determine them for the second, third, &c. ascents. Let E a' represent the height of the water after the second ascent, and let it = x1, the height due to the, elastic force of the air; then x1+ y1 = h; Ab Να = whence h and .. x1 = 1 since the air which occupied ca now occu 1. {p. and y1 = . {2h-p+ √ p2—4 h m b—4 x . (h+a — x)}. 1. From these are deduced values of x, y, x3, Y 39 -4 h m b-4 x1. (h+a — x1)}, and so on. Whence if x, be taken to represent the height of the water after (n + 1) ascents, and y..{2h-p+ √p2-4 h m b-4x-1. (h+a−x-1)}. |