the mantissa, the product has a characteristic, add it, algebraically, to the negative characteristic, multiplied by the exponent, and the result will be the negative characteristic of the required power. Art. 658. EVOLUTION BY LOGARITHMS. Rule.—To extract any root of a number by means of logarithms, divide the number by the index of the root; the result will be the logarithm of the root. Art. 662. If it is required to extract a root of a number wholly decimal, and the negative characteristic will not exactly contain the index of the root, without a remainder, proceed as follows: Separate the two parts of the logarithm; add as many units (or parts of a unit) to the negative characteristic as will make it exactly contain the index of the root. Add the same number to the mantissa, and divide both parts by the index. The result will be the characteristic and mantissa of the root. Art. 663. FORMULAS USED IN ELEMENTARY UNIFORM MOTION. = Let S the length of space passed over uniformly; the time occupied in passing over the space S; V the velocity. t = MASS, WEight, and GRAVITY. If the mass of the body be represented by m, its weight by W, and the force of gravity at the place where the body was weighed by g, we have = Let W weight of body at the surface; W = weight of a body at a given distance above or below the surface; d = distance between the center of the earth and the center of the body; Formula for weight when the body is below the surface: w Rd W (11.) Art. 891. Formula for weight when the body is above the surface. Let g force of gravity = force of gravity = constant accelerating force due to the attraction of the earth; t = number of seconds the body falls; v = velocity at the end of the time t; h = distance that a body falls during the time t. That is, the velocity acquired by a freely falling body at the end of t seconds equals 32.16 multiplied by the time in seconds. That is, the number of seconds during which a body must have fallen to acquire a given velocity equals the given velocity in feet per second divided by 32.16. That is, the height from which a body must fall to acquire a given velocity equals the square of the given velocity divided by 2 × 32.16. That is, the velocity that a body will acquire in falling through a given height equals the square root of the product of twice 32.16 and the given height. h = gť. (17.) Art. 896. That is, the distance a body will fall in a given time equals 32.16÷2 multiplied by the square of the number of seconds. That is, the time it will take a body to fall through a given height equals the square root of twice the height divided by 32.16. CENTRIFUGAL FORCE. The value of the centrifugal force of any revolving body, expressed in pounds, is F.00034 WR N2; in which F centrifugal force; = (19.) Art. 903. W = total weight of body in pounds; R = radius, usually taken as the distance between the center of motion and the center of gravity of the revolving body, in feet; N number of revolutions per minute. THE CENTER OF GRAVITY OF TWO BODIES. Let / 1, = = W = the distance between the centers of the bodies; the short arm; w weight of small body; W = weight of large body. Let F THE EFFICIENCY OF A MACHINE. the force applied to the machine; V = the velocity ratio of the machine; W = the weight actually lifted or equivalent resist If the force necessary to overcome the resistance be represented by F, the space through which the resistance acts by S, and the work done by U, then UF S. h If W the weight of a body, and the height through which it is raised, U= Wh. Hence the work done U=FS= Wh. (23.) Art. 953. POWER. The power of a machine may always be determined by dividing the work done in foot-pounds by the time in minutes required to do the work; i. e., Let W the weight of the body in pounds; vits velocity in feet per second; the height in feet through which the body must fall to produce the velocity v; W m = the mass of the body = (See formula 10.) g The work necessary to raise a body through a height his Wh. The velocity produced in falling a height his Let D equal the density, m the mass, and W the weight, and the volume (in cu. ft.) of a body. Then PRESSURE, VOLUME, DENSITY, AND WEIGHT OF AIR Mariotte's Law.-The temperature remaining the same, the volume of a given quantity of gas varies inversely as the pressure. Let p = pressure for one position of the piston; P1 pressure for any other position of the piston; = volume corresponding to the pressure p; volume corresponding to the pressure p1. = Then, pvp12 (53.) Art. 1049. Let D be the density corresponding to the pressure and volume v, and D, be the density corresponding to the pressure, and volume v,; then, whose volume is v, and pressure is p; let W, be the weight of a cubic foot when the volume is 7,, and pressure is p1; then, |