MATHEMATICAL THEORY.] AERONAUTICS 203 caught fire and was burnt. lu 1871 a series of experi ments was made at Penn s factory (Greenwich) on the resistance of different shaped planes placed at different angles, in a current of air produced by a rotary fan. In vestigations of this kind not only form the first step towards obtaining data for a true knowledge of the exact nature of flying, but are also independently of high scientific interest. The chief object of the society is to bring together those persons who are interested in the subject of aero nautics (except balloonists by trade, who are ineligible), and to encourage those who, possessing suitable acquire ments, are devoting their time to the investigation of the question. Aerostatic societies have also been founded in other countries; but although they have been inaugurated with considerable eclat, more than one have already terminated a short-lived career. The Vienna society seems, however, to have been unusually active during the recent exhibi tion of 1873. >ry of The principle in virtue of which a balloon ascends is quili- exactly the same as that which causes a piece of wood or n n f _ other material to float partially immersed in water, and " may be stated as follows, viz., that if any body float in equilibrium in a fluid, the weight of the body is equal to the weight of the fluid displaced. By the " fluid dis placed " is meant the fluid which would occupy the space actually occupied in the fluid by the body if the body were removed. When the fluid is inelastic and incompressible, i.e., a liquid, as water, its density is the same throughout, and bodies placed in it either rise to the surface and float there partially immersed, or sink to the bottom. Thus, suppose a body only one-third as heavy as water (in other words, whose specific gravity is one-third) was floating on the surface of water, then, as the weight of the body must be equal to that of the water it displaces, it is clear that one- third of the body must be immersed. In the case, however, of an elastic or gaseous fluid, such as air, the density gradu ally decreases as we recede from the surface of the earth, for each layer has to support the weight of all above it, and as air is elastic or compressible, the layers near the earth are more pressed upon, and therefore denser than those above. Thus, if a body lighter than the air it displaces be set free in the atmosphere, it rises to such a height that the air there is so attenuated that the weight of it displaced is equal to that of the body, when equilibrium takes place, and the body ascends no higher. In all cases, therefore, a body floating in the air is totally immersed, and it can never get beyond the atmosphere, and float, as it were, upon its surface. To find, therefore, how high any body (lighter than the air it displaces), such as a balloon, of given capacity and weight, will rise, it is only necessary to calculate at what height the volume of a quantity of air equal to the given capacity will be equal in weight to the given weight. Leaving temperature out of the question, the law of the decrease of density in the atmosphere is such that the ^ density at a height x is equal to e * * x the density at the earth s surface, g being the measure of gravity, and Jc also k a constant; the value of - is called the height of the homo geneous atmosphere, viz., it is equal to what would be the height of the atmosphere if it were homogeneous through out, and of the same density as at the earth s surface. Its value may be taken at about 26,000 feet. Thus, let V be the volume of a balloon and its appurtenances, car, ropes, &c. (viz., the number of cubic feet, or whatever the unit of solidity may be, that it displaces), and let W be its weight (including that of the gas), then it will rise to a height x such that W = x density of air, g being the value of the force of gravity, and cr being the density of the air at the surface of the earth. This equa tion is not quite accurate, for several reasons (1) because the decrease of temperature that results from increase of elevation has not been taken into account; (2) because g has been taken to measure the force of gravity on the earth s surface, whereas it should represent this force at a height x; this is easily corrected by replacing g by g f where g =g- - , a being the radius of the earth, but ci -f- x) as a is about 4000 miles, and x is never likely in any ordi nary question to exceed 1 miles, we can replace g by g without introducing sensible error, for the correction due to this cause would be much less than other uncertainties that must arise; and (3) because Wand V could not both remain constant. If the balloon be not fully inflated on leaving, so that the gas contained in it can expand, then V, the volume of air displaced, will increase; while, if the balloon be full at starting, the envelope must either be strong enough to resist the increased pressure of the gas inside, due to the removal of some of the pressure outside (owing to the diminished density of the air), or some of the gas must be allowed to escape. The former alternative of the second case could not be complied with, as the balloon would burst; some of the gas must therefore escape, and so W is diminished. The weight of gas of which the balloon is thus eased cannot properly be omitted from the calculation, if x be considerable ; but a good approximation is obtained without it, as the weight of the gas that escapes will generally bear a small proportion to the weight of balloon, car, grapnel, passengers, &c. The true equation (except as regards temperature) is therefore, for a balloon full at starting > / _ fT - ,., od"t fl p (l e~ t *) gQi (t e * V . L-iJ = " v denoting the volume actually occupied by the gas, g* denoting ^f- - , viz., gravity at height x, and p u being (ft T X) the density of the gas on the ground. It will generally be sufficient, especially when temperature is omitted, to take the formula in the approximate form written previously. As the volume of air displaced by the car, ropes, passengers, &c., is usually trifling compared to that displaced by the balloon itself, no great error can arise from taking ^ = Y . As an example, let us find how high a balloon of 100,000 cubic feet capacity would rise if inflated with pure hydrogen gas, carrying with it a weight of 3000 Ib (this including the weight of the balloon itself and appurtenances). A cubic foot of air, at temperature 32 Fahr., and under a pressure of 29 922 in., weighs 080728 Ib, and a cubic foot of hydrogen weighs 005592 ft, so that (supposing the barometer reading on the earth to be 29 922 in., and the temperature of the air to be 32) at the surface of the earth the balloon, &c., weighs 3559 Ib, and the weight of the air displaced is S073 ft. The balloon will therefore approximately rise to such a height x that 100,000 cubic feet of air shall there weigh 3559 ft; and x is given in feet by the equation 3559 6 "*"- 8073* or x = 26,000 (log 8073 - log 3559), the logarithms being hyperbolic; if common or Briggian logarithms be used, the result must be multiplied by
2 30258 . . . (the reciprocal of the modulus). In the above
