Mathematical Theory of the Motion of the Balloon
The mathematical theory of the rate of ascent of a balloon 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 investigate the motion of a globe lighter than the air it displaced. For 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 earostat. This investigation is printed under the title, Calculs sur les Ballons Aérostatiques faits par feu. M. Léonard Euler, tells qu on les a trouvés sur son ardoise, après sa mort arrivée le 7 Septembre 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 Montgolfiers were made with balloons inflated with heated air. The explanation of this must be that either an imperfect accounts 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 he balloon's motion a peculiar interest.
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, ad 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 transform 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 allowable 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 observation, that the rate of the decline of temperature when the sky is cleat 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., we have supposed the temperature to remain constant throughout the atmosphere. The assumption of any law of decrease would considerably complicate the equations. Perhaps the simplest law, mathematically considered, would be to assume the curve of descent of temperature to be y = e --ax. The curve Mr. Glaisher deduced from his eight ascents was a portion of a hyperbola, the constants being determine empirically.
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