204 AERONAUTICS [MATHEMATICAL athema-
- al theov
the otion of balloon
- e last
obi em .at en- iged the tention Euler. >tion halloo; case we find x = about 21,000 feet, and as at this height rather more than half the gas will have escaped (it having been supposed that the balloon was full at starting). This only reduces the value 3559 by about 300, and the result of taking it into account is only to increase the height just found by about 200 feet. If 2000 R> out of the 3000 were thrown away during the ascent, the balloon would reach a height of about 10 miles; the weight of the gas that escapes is here important, as, if it be not taken into account, the height given by the formula is only about 8 miles. In actual aerostation, as at present practised, ordinary coal gas is used, which is many times heavier than hydrogen, being, in fact, usually not less than half the specific gravity of air. Even when balloons are inflated with hydrogen, generated by the action of sulphuric acid on zinc filings, the gas is very far from pure, and its density is often double that of pure hydrogen, and even greater. The hydrostatic laws relating to the equilibrium of float ing bodies were known long previous to the invention of the balloon in 1783, but it was only in the latter half of the 18th century that the nature of gases was sufficiently understood to enable these principles to have been acted on.- As we have seen, both Black and Cavallo did make use of them on a small scale, and if they had thought it possible to make a varnish impervious to the passage of hydrogen gas they could have easily anticipated the Montgolfiers. As it was, no sooner was the fire-balloon invented, than Charles at once suggested and practically carried out the idea of the hydrogen or inflammable air balloon. The mathematical theory of the rate of ascent of a oalloon possesses remarkable historic interest, from the fact that it was the last problem that engaged the attention . of the greatest mathematician of the last century, Euler. The news of the experiment of the Montgolfiers at Annonay on June 5, 1783, reached the aged mathematician (he was in his 77th year) at St Petersburg, and with an energy that was characteristic of him he at once proceeded to investi gate the motion of a globe lighter than the air it displaced. Eor many years he had been all but totally blind, and was in the habit of performing his calculations with chalk upon a, black board. It was after his death, on September 7, 1783, that this board was found covered with the analytical investigation of the motion of an aerostat. This investi gation is printed under the title, Calculs sur les Ballons Aerostatiques fails par feu M. Leonard Euler, ids qu on les a trouves sur son ardoise, apres sa mort arrivee le 7 Septemhre 1783, in the Memoirs of the French Academy for 1781 (pp. 264-268). The explanation of the earlier date is that the volume of memoirs for 1781 was not published till 1784. The peculiarity of Euler s memoir is that it deals with the motion of a closed globe filled with a gas lighter than air, whereas the experiments of the Mont golfiers were made with balloons inflated with heated air. .The explanation of this must be that either an imperfect account reached Euler, and that he supplied the details himself as seemed to him most probable, or that he, like the Montgolfiers themselves, attributed the rising of the balloon to the generation of a special gas given off by the chopped straw with which the fire was fed. The treatment of the question by Euler presents no particular point of importance indeed, it could not; but the fact of its having .given rise to the closing work of so long and distinguished a life, and having occupied the last thoughts of so great a mind, confers on the problem of the balloon s motion a peculiar interest. of We now proceed to the investigation of the vertical . motion of a balloon inflated with gas, the horizontal motion, of course, being always equal to that of the current in which it is placed. In supposing, therefore, the balloon to be ascending vertically into a perfectly calm atmosphere, there is no loss of generality. There are two cases of the problem, viz., when the balloon is only partially filled with gas at starting, and when it is quite filled. The motion in the former case we shall investigate first, as the balloon will ascend till it becomes completely full, and then the subsequent motion will belong to the second case. We may remark that it is usual in investigations relating to the motions of a balloon to regard it in the way that Euler did, viz., as a closed inextensible bag, capable of bearing any amount of pressure. In point of fact, the neck or lower orifice of the balloon is invariably open while it is in the air, so that the pressure inside and outside is practically always the same, and when the balloon continues ascending after it has become quite full, the gas pours out of the neck or is allowed to escape by opening the upper valve. It is to be noticed that we have not thought it necessary to trans form the formulae obtained in such wise that they may be readily adapted to numerical calculations as they stand, as our object is rather to exhibit the nature of the motion, and clearly express the conditions that are fulfilled in the case of a balloon, than deduce a series of formulae for practical use. We shall, however, indicate the simplifications allow able in practical applications. The effect of temperature, though important, is neglected, as the connection between it and height is still unknown. It was chiefly to determine this relation that Mr Glaisher s ascents were undertaken, and at the conclusion of the first eight he deduced an empirical law which seemed to accord pretty well with the observations ; the succeeding twenty ascents, however, failed to confirm this law. In fact, it is evident, even without ob servation, that the rate of the decline of temperature when the sky is clear must differ from what it is when cloudy, and that, being influenced to a great amount by radiation of heat from the earth s surface, it will vary from hour to hour. Under these circumstances, as our object is not to deduce a series of practical rules for calculating heights, &c., Ave have supposed the temperature to remain constant throughout the atmosphere. The assumption of any law of decrease would considerably complicate the equations. Per haps the simplest law, mathematically considered, would be to assume the curve of descent of temperature to be y = e~ aa . The curve Mr Glaisher deduced from his eight ascents was a portion of a hyperbola, the constants being determined empirically. Let M = the mass of the balloon, car, netting, gas, pas- Motion sengers, &c., on starting. a halloo V = the capacity of the envelope of the balloon when only pa full. tially i r s = the volume of gas at the pressure of the air iiitro- at start: cluced into the balloon before starting. v = the volume (supposed less than V ) occupied by the gas at the height x. O Q = density of the gas in the balloon on the earth. e = at the height i7 = density of the air on the earth. ff - at the height x. u = the initial upward velocity of the balloon (which 13 introduced for the sake of complete generality, but is always zero). M = the velocity (vertically upwards, as all horizontal motion is ignored) at height x. Then the equation of motion at any time previous to the balloon becoming completely filled is nr ll , MM = ovg - / 9 * * r - Awe * , the last term being due to the resistance of the air, which is assumed to vary directly as the square of the velocity
and as the density of the air. In very slow motions the
