AERONAUTICS (Part 3)
The paramount value of the balloon during the recent siege of Paris must be fresh in the minds of all. It was by it alone that communication was kept up between the besieged city and the external world, as the balloons carried away from Paris the pigeons which afterwards brought back to it the news of the provinces. The total number of balloons that ascended from Paris during the siege, conveying persons and dispatches, was sixty-four-the first having started on September 23, 1870, and the last on January 28, 1871. Gambetta effected his escape from Paris, on October 7, in the balloon Armand-Barbes, an event which doubtless led to the prolongation of the war. Of the sixty-four balloons only two were never heard of;' they were blown out to sea. One of the most remarkable voyages was that of the Ville d'Orleans, which, leaving Paris at eleven o'clock on November 21, descended fifteen hours afterwards near Christiania, having crossed the North Sea. Several of the balloons on their descent were taken by the Prussians, and a good many were fired at while in the air; but we do not hear of any being injured from this cause. The average size of the balloons was from 2000 to 2050 metres, or from 70,000 to 72,000 cubic feet. The above facts we have extracted from Les Balloons du Siege de Paris, a sheet published by Bulla & Sons, Paris; compiled by the brothers Tissandier, well-known French aeronauts, and giving the name, size, and times of ascent and descent of every balloon that left Paris, with the names of the aeronaut and generally also those of the passengers, the weight of dispatches, the number of pigeons, &c. Only those balloons, however, are noticed in which some person ascended. A similar list of sixty-twp balloons is given by Mr. Glaisher in the introduction to the second edition of Travels in the Air (1871). It was, however, published too soon after the conclusion of the siege to be quite so complete as the sheet of the MM. Tissandier. It is perhaps worth stating that the balloons were manufactured and dispatched (generally from the platforms of the Orleans or the Northern Railway) under the direction of the Post-Office. The aeronauts employed were mostly sailors, who did their work very well. No use whatever was made in the war of balloons for purposes of reconnaissance. The exceedingly important part played by the balloon in the siege of Paris would alone, if it had been of no other utility, render it one of the most valuable inventions of the last century.
The principle of the parachute is so simple that the idea must have occurred to persons in all ages. Father Loubere in his History of Siam, published two centuries ago, tells of a person who frequently diverted the court by the prodigions leaps he used to take, having two parachutes or umbrellas fastened to his girdle. In 1783 a certain M. le Normand practically demonstrated the efficiency of a parachute by descending from a high house at Lyons; but he merely regarded it as a useful means whereby to escape from fire. To Blanchard is due the idea of using it as an adjunct to the balloons. As early as 1785 he had constructed a parachute, to which was attached a basket. In this he placed a dog, which descended safely to the ground when the parachute was released from a balloon at a considerable elevation. It is stated that he descended himself from a balloon in a parachute in 1793; but owing to some defect in its construction, he fell too rapidly, and broke his leg.
Andres Jaques Garnerin was the first person who successfully descended from a balloon in a parachute, and he repeated this experiment so often that he may be said to have first demonstrated the practicability of using the machine; and, in fact, that he invented it in a practical and suitable form. In 1793 Garnerin had been taken prisoner at Marchiennes, and he was confined for between two and three years in the fortress of Bude, in Hungary. While in captivity he elaborated in his mind the means of descending from a balloon by means of a parachute; and on October 22, 1797, he made his first public experiment. He ascended from the park of Moncau at Paris, and when at the height of about 1 x mile he released the parachute, which was attached to the balloon in place of a car; the balloon, relieved suddenly of so great a weight, rose very rapidly till it burst, while the parachute ddescended very fast, making violent oscillations all the way. Garnerin, however, reached the earth in safety upon the pain of Monceau. In 1802 Garnerin came to England and made a good many ascents in all parts of the country, many of which excited much enthusiasm, as can be seen from the contemporary accounts; and o September 21, 1802, he repeated his parachute experiment in England .
The parachute was done-shaped, and bore a resemblance to a large umbrella. The case or dome was made of white canvass, and was 23 feet in diameter. At the top was a truck or round piece of wood 10 inches in diameter, with a hole in its center, fastened to the canvass by 32 short pieces of tape. The parachute was suspended from a hoop attached to the netting of the balloon, and below the parachute was placed a cylindrical basket, 4 feet high and 2 x feet in diamter, which contained the aeronaut. The ascent took place at about six o'clock from North Street, London; and, at a height of about (it is believed) 8000 feet, Garnerin separated the parachute from the balloon. For a few seconds his fate seemed certain, as the parachute retained the collapsed state in which it had originally ascended, and fell very rapidly. It suddenly, however, expanded, and the rapidity of its descent was at once checked, but the oscillations were so violent that the car, which was suspended 20 feet below, was sometimes on a level with the rest of the apparatus. Some accounts state that these oscillations increased, others that they decreased as the parachute descended, and the latter seems most probable. It came to the ground in a field at the back of St Pancras church, the descent having occupied rather more than ten minutes. Garnerin was hurt a little by the violence with which the basket containing him struck the earth; but a few cuts and a slight nausea represented all the ill effects of his fall. He made, certainly, one other descent in a similar was (as that just described is stated to have been his third), and we believe several others on the Continent, but this was the only on he effected in England.
Jordaki Kuparento, a Polish aeronaut, is the only person who ever made any real use a parachute. He ascended from Warsaw on July 24, 1808, in a fire-balloon, which, at a considerable elevation, took fire; but being provided with a parachute, he was enabled to effect his descent in safety.
The next experiment made with a parachute was that which resulted in the unfortunate death of Mr. Robert Cocking. So early as 1814 this gentleman had lectured on the subject before the City Philosophical Society, and also before the Society of Arts. He always retained an interest in ballooning, and made two ascents -- one with Mr, Sadler, and the other on September 27, 1836, with Mr. Green. The success of the balloon trip of Messrs Hollond, Mason, and Green, seems to have incited Mr. Cocking to demonstrate practically the truth of his views. He accordingly constructed a suitable parachute on his principles, and having succeeded in obtaining the consent Messrs Hughes and Gye, the proprietors of Vauxhall Gardens, to permit the ascent to be made there, he prevailed on Mr. Green to ascend in his great Nassau balloons with the parachute attached. The great defect of Garnerin's umbrella-shaped parachute was its violent oscillation during descent, and Mr. Cocking considered that if the parachute were made of a conical (vertex downwards), the whole of this oscillation would be avoided; and if it were made of sufficient size, there would be resistance enough to check too rapid a descent. He therefore constructed a parachute on this principle, the radius of which at is widest part was about 17 feet, it was stated in the public announcement previous to the experiment that the whole weighted 223 lb; but from the evidence at the inquest it appeared that the weight must have been over 400 lb. Mr. Cocking's weight was 177 lb. Which was so much additional. On July 24, 1837, the trial took place; and the Nassau balloon, with Mr. Green and Mr. Spencer, a solicitor, in the car, and having suspended below it the parachute, in the car of which was Mr. Cocking, rose from the ground at twenty-five minutes to eight in the evening. A good deal of difficulty was experienced in rising to a suitable height, partly in consequence of the resistance to the air offered by the expanded parachute, and partly owing to its weight. Mr. Cocking wished the height to be 8000 feet; but when the balloon reached the height of 5000 feet, it being then nearly over Greenwich, Mr. Green called out to Mr. Cocking that he should be unable to ascended to the requisite height if the parachute was to descend in daylight. Mr. Cocking accordingly let ship the catch which was to liberate him from the balloon. The parachute for a few seconds descended very rapidly but still evenly, until suddenly the uppr rim seemed to give way, and thre whole apparatus collapsed )taking a form resembling an umbrella turned inside out, and nearly closed), and the machine descended with great rapidity; oscillating very much. When about two or three hundred feet from the ground, the basket became disengaged from the remnant of the parachute, and Mr. Cocking was found in a field at Lee, literally dashed to pieces.
Mr. Green and Mr. Spencer, who were in the car of the balloon, had also a narrow escape. At the moment the parachute was disengaged they crouched down in the car, and Mr. Gren clung to the valve-line, to permit the escape of the gas. The balloon shot upwards, plunging and rolling, and the gas pouring both the upper and lower valves, but chiefly from the latter, as the great resistance of the air checked its egress from the former. Mr. Green and Mr. Spencer applied their mouths to tubes communicating with an air bag with which they had had the foresight to provide themselves, otherwise they would certainly have been suffocated by the gas. Notwithstanding his precaution, however, the gas almost totally deprived them of sight for four or five minutes. When the came to themselves they found they were at a height of about four miles, and descending rapidly. They effected, however, a safe descent near Maidstone.
Many objections were made, after the result, to the form of Mr. Cocking's parachute; but there is little doubt that had it been constructed of sufficient strength, and perhaps of somewhat larger size, it would have answered its purpose.. as it was, the upper rim was made of tin, which soon gave way. Mr. Wise, the American aeronaut, made some experiments on parachutes of both forms (Garnerin's and Cocking's), and found that the latter always were much more steady, descending generally in a spiral curve.
In 1839 Mr. Hampton made three descents in a parachute, on Garnerin's pattern, from his balloon, the "Albion." He followed Garnerin's example in attaching the parachute to the netting of the balloon, so that when between the two was severed the latter was left to its own devices. Mr. Hampton took measures, however, that it should descend soon after the parachute, and it was generally found no great distance off, and returned to him. All his parachute descents were safely performed, although in one he was a good deal shaken.
We may remark that a descending balloon half-full of has either does rise, or can with a little management be made to rise, to the top of the netting and take the form of a parachute, thus materially lessening the rapidity of descent. Mr. Wise, in fact, having noticed this, once purposely exploded his balloon when at a considerable the altitude, and the resistance offered to the air by the envelope of the balloon was sufficient to enable him to reach the ground without injury. And a similar thing took place in one of Mr. Glaisher's high scientific ascent (April 18, 1863), when, at a height of about 2 miles, the sea appeared directly, underneath; the gas was let out of the balloon as quickly as possible, and the velocity of descent was so great, that the 2 miles of vertical height were passed through in four minutes. On the balloon reaching the ground at Newhaven, close to the shore, it was found to be nearly empty. The balloon had, in fact, for the last mile or more, merely acted as a parachute; the shock was a severe one, and all the instrument were broken, but nothing serious resulted to the occupants of the car.
Numerous attempts have been made both to direct balloons and contrive independent flying machines. After the invention of the balloon by the brothers Montgolfier, it was at once thought that no very great difficulty would be found in devising a suitable steering apparatus; in fact, it was supposed that to rise into the air and remain thre was the chief difficulty, and that, this being accomplished, the power of difrecting the aerostat would be secondary achievement that must follow before long. Accordingly in most of the early balloons the voyages took up oars, sails, or paddles, which they diligently worked while in the air; sometimes they thought an effect was produced, and sometimes not. If we consider the number of different currents in the atmosphere, it is no wonder that some should have announced with confidence that their course was changed from that of the wind by means of the sails or oars that they used; in fact, it is not very often that the whole atmosphere up to a considerable height is moving en masse in the same direction, so that generally the course taken by the balloon, as determined merely joining the places of ascent and descent, is not identical with the direction of the wind, even when it is the same at both places. Although there is no reason why balloons should not be so guided by means of mechanical appliances attached to them as to move in a direction making a small angle with that of the wind, still it must have been evident to any one who has observed a balloon during inflation on a windy day, that any motion in which it would be exposed to the action of a strong current of air must result in its destruction. It has therefore gradually become recognized that the balloon is scarely a step at all towards a system of aerial navigation; and many have thought that the principles involved in the construction of a flying machine must be very different from the simple statical equilibrium that subsists when a balloon is floating in the air. "To navigate the air the machine must be heavier than the air,." Has frequently been regarded as an axiom; and there can be no doubt that an apparatus constructed of such light material as is necessary for a balloon either be destroyed or become ungovernable in a high wind. Recently, however, M. Dupuy de Lome, an eminent French engineer, gas constructed and made experiments with a balloons which he considers satisfies some of the conditions. The balloon is spindle-shaped, the longer axis being horizontal, and it contains about 12,000 cubic feet. The car is suspended below the middle of the balloon, and there are provided a rudder and a screw. The rudder consists of a triangular sail placed beneth the balloon and near the rear, and is kept in position by a horizontal yard, about 20 feet long, turning round a pivot in its forward extremity; the height of the sail is 16 feet, and its surface 160 square feet. Two ropes for working the rudder extend forward to the seat of the steerer, who has before him a compass fixed to the car, the central part of which will contain fourteen men. The screw is carried by the car, and is driven by four or eight men working at a capstan. A trial was made with the machine on Febrary 2, 1872, on a windy day, and M. de Lome considered that he had been enabled by his screw and rudder to alter his course about 12°. (See Report of the Aeronautical Society, 1872).
Whatever difficulties may present themselves in regulating the horizontal movement of the balloon, there can be no doubt that the vertical motion could be obtained by means of a screw or other mechanical means; and the power of being able to ascend or descend without loss of ballast would be a considerable gain. In the opinion of many, however, the balloon is not worth improvements; and as ballooning is now generally practiced merely as a spectacle by which the aeroanut or showman gains his living, it is not likely that any advancement will be made.
Of flying machines, in which both buoyancy and motion were proposed to be obtained by purely mechanical means the number has been very great. Most of the projects have been chimerical, and were due to persons possessed of an insufficient knowledge of the principles of natural philosophy, both theoretically and practically. They serve, however, to show how great a number of individuals must have said attention to the matter, and even at the present time several patents are taken out annually on the subject. We do not propose her to give an account of any of these projects, fort but few have ever passed beyond projects, but will merely refer to Mr. Henson's aerial carriage, which in 1843 attracted some attention. The apparatus was an elaborate one, and its principal feature was the great expanse of the sustaining planes. The machine was to advance with its front edge a little raised, the effect of which would be to present its under surface to the air over which it was passing; the resistance of this air, acting on it like the strong wind on the sails of a wind will, would, it was thought, prevent the descent of the machine. Mr. Henson invented a steam-engine of great lightness, but he proposed that the machine should be started down an inclined plane, so that the steam-engine would only have to make up for the velocity lost by the resistance of the air. The scheme never came to anything.
In the still air of a room it is, of course, not difficult to attach an apparatus to a balloon so as to direct its motion, and even models of flying machines have been which, when tried in a room, seemed moderately successful. Some instruments which would vry nearly support themselves in the air were shown at the Aeronautical Society's exhibition at the Crystal Palace. A good deal would be accomplished if an accurate knowledge of the exact motion of a bird's wing could be obtained; in fact, until this is known, or until sufficient experiments on the resistance experienced by different-shaped lamine with different motion are made, there seems little chance of the construction of a satisfactory flying machine, unless means can be found to make a stream-engine of much less weight than is at present necessary.
In 1865 the Aeromautical Society of Great Britain was founded, the officers being-President, the duke of Argyle Treasurer, Mr. J. Glaisher; and Secretary, Mr. Brearey. It has published an annual report every year since (1873_, containing selections from the papers read to the society, and abstracts of the discussions that took place thereon at the meetings. The numerous papers submitted to this society bear witness to the great number of minds that are engaged on the solution of the problem of aerial navigation. Of course, not a few of the methods proposed are the fanciful projects of ignorant men, but some show the careful though and elaborate experiment of trained engineers and other qualified persons. In 1868 the society held an exhibition of flying machines, &c., at the Crystal Palace, which was visited by many persons. A fire-balloon of a M. de la Marne, which should have ascended during his exhibition, caught fire and was burnt. In 1871 a series of experiments 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. Investigations 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 aeronautics (except balloonists by trade, who are ineligible), and to encourage those who, possessing suitabl acquirements, 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 xclat, more than one have already terminated a short-lived career. The Vienna society seems, however, to have been unusually active during the recent exhibition of 1873.
The principle in virtue of which a balloon ascends is exactly the same as that which causes a piece of wood or 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 Îfluid displaced" 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 gradually 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 FORMULA x the density at the earth's surface, g being the measure of gravity, and k also a constant; the value of FORMULA is called the height of the homogeneous atmosphere, viz, it is equal to what would be the height of the atmosphere if it were homogeneous throughout, 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 displaced), and let W be its weight (including that of the gas), then it will rise to a height x such that
FORMULA
G being the value of the force of gravity, and xo being the density of the air at the surface of he earth. This equation is not quite accurate, for several reasons -- (1) because the decrease of temperatures that results from increase of elevation has not been taken into account; (2) because of 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, where of = FORMULA a being the radius of the earth, but as a is about 4000 miles, and x is never likely in any ordinary question to exceed 10 miles, we can replaced 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 W and 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 to resist the increased pressure of the gas inside, due to 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-
FORMULA
vo denoting the volume actually occupied by the gas, g denoting FORMULA viz., gravity at height x, and po being 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 vo = Vo. As an example, let us find now high a balloon of 1000,000 cubic feet capacity would rise if inflated with pure hydrogen gas, carrying with it a weight of 3000 lb (this including the weight of the balloon itself and appurtenances). A cubic foot of air, at temperature 32o Fahr., and under a pressure of 29.922 in., weighs 080728 lb, and a cubic foot of hydrogen weighs 005592 lb, so that 9supposing the barometer reading on the earth to be 29.922, and the temperature of the air to 32o) at the surface of the earth the balloon, &c., weighs 3559 lb, and the weight of the air displaced is 8073 lb. The balloon will therefore approximately rise to such a height x that 100,000 cubic feet of air shall there weigh 3559 lb; and x is given in feet by the equation.
FORMULA
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 case we find x = about 21,000 feet, and a 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 lb 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 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 practiced, 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 floating 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 too 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 gas they could have easily anticipated the Montgolfiers. As it was, so 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 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 Montgolfies at Annonay on June 5, 1783, reached the aged mathematician (he was in his 77th year) at St Petersburgh, 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 Aerostatiques faits par feu. M. Leonard Euler, tells qu on les a trouves sur on ardoise, aprxs sa mort arrivee 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.
-Let M = the mass of the balloons, car, netting, gas, passengers, &c., on starting.
vo = the capacity of the envelope of the balloon when full.
vo= the volume of gas at the pressure of the air introduced into the balloon before starting.
v= he volume (supposed less than Vo ) occupied by the gas at the height x.
po = density of the gas in the balloon on the earth.
p= density of the gas in the balloon at the height x.
xo = density of the air on the earth.
x = density of the air at the height x.
u= the initial upward velocity of the balloon (which is introduced for the sake of complete generality, but is always zero).
uo = 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
FORMULA
The last term being due to the resistance of the air, which is assumed to vary directly as the square of he velocity and as the density of the air. In very slow motions these resistance appears from experiments to vary pretty nearly as the velocity; and when the motion is very swift, as in the case of a rifle-bullet, as the cube of the velocity; but when the motion is neither very rapid nor very slow, the law of the square of the velocity probably represents the truth very fairly. By g is denoted the value of gravity at the height x, so that
FORMULA
a being, as above, the radius of the earth. In the exponential term, we shall replace g by g, as no sensible error can result thereform. The value of xu is constant, as by Boyle's and Marriotte's law it always = xo vo. Writing, therefore, for brevity --
FORMULA
The equation of motion rakes the form
FORMULA
Whence, following the usual rule for the integration of linear differential equations of the first other, and writing X for e-nz, for convenience of printing,
FORMULA
Herein put x = 0, so that u = uo and we have
FORMULA
Whence, by subtraction,
FORMULA
Therefore
FORMULA
In which Ei x I used to denote the exponential integral of z, viz.: FORMULA according to a recognized notation. The values of the integral Ei x, which may be regarded as a known function, have been tabulated (see Philosophical Transactions for 1870, pp. 367-388).
We thus have, except for temperature, the complete solution of the problem of the motion of the balloon so far as velocity and height are concerned; it would not be possible to connect the time and the height except by the performance of another integration, for the practicability of which it would be necessary to submit to some loss of generality, viz., we should have to regard x as small as compared to a, and take x as small, and so on. The equation last written gives the motion until the height (say h) is attained at which the balloon, becomes quite full, after which the gas begins to escape, and have the second case of the problem.
Before proceeding, however, to the discussion of this second case, it is worth while to examine the solution more carefully, leaving out of consideration quantities that make no very great difference in the practical result, for the sake of simplicity. Supposing, then, gravity to be constant at all heights, and x to be zero, the equation of motion takes the simple form.
FORMULA
Whence
And we see, what is pretty evident from general reasoning, that if a balloon, partially filled, rises at all, it will at least rise to such a height that it will become completely full.
The letters meaning the same as before, the equation of motion of a balloon completely filled at starting is
FORMULA
Or substituting for p and x their values
FORMULA
The integral of this differential equation could be obtained in series as before, only that the resulting equations would be more complicated. As we do not propose to discuss the formulae obtained, it will be sufficient for our purpose to deduce an approximate solution by neglecting Vo po (1 -- e --nz) compared to M, viz neglecting the mass of the gas that has escaped during the ascent compared to the mass of the whole balloon and appurtenances. It must be borne in mind, however, that when coal gas is used, and the ascent is to a great height, the mass of gas that escapes is by no means insensible. The equation thus becomes
FORMULA
x being FORMULA. This is an equation which can be integrated in exactly the same was as that previously considered, viz, by multiplying by a factor e-mx, and integrating at once; thus,
FORMULA
FORMULA
And C is determined as before by putting x = 0, when we have u = uo.
In this case uo is not zero, except when the balloon starts from the earth quite full. The general case is when the balloon is only partially filled on leaving; the previous equations then hold until a height h, at which it becomes quite full, when the motion changes, and is as just investigated. Then uo becomes the velocity at the height, h, and everything is measured from this height as if from the surface of the earth, a being then the radius of the earth + h, po, xo the densities at height, h,m and p, x at height x + h, &c. We have therefore, except as regards time, completely determined the motion of a balloon inflated with gas in an atmosphere of constant temperature. The introduction of temperature would modify the motion considerably, but in the present state of science it cannot be taken into account.
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The general principle of the equilibrium of a fire-balloon is, of course, identical with that of a gas-balloon, but the motion is different, as the degree of buoyancy at each moment varies with the temperature of the air within the balloon, and therefore with the heat of the furnace by which the air is warmed. Dry air expands FORMULA part of its volume for every increase of temperature of 1o centigrade, or FORMULA of its volume for very increase of temperature of 1° Fahr. More than the surrounding air, the air within the bag will expand FORMULA of its volume, and this air must therefore escape. The air within the bag weighs less, therefore, than the air it displaces by the FORMULA part of the latter; and if the weight of this de greater than the weight of the bag and appurtenances, the latter will ascend. It is, therefore, always easy to calculate approximately the ascensional power of a fire-balloon if the temperature of the surrounding air be known, and also the mean temperature of the air within the balloon. Thus, let the balloon contain V cubic feet of hot air at the temperature t (fahr.), and let the temperature of the surrounding air be t (Fahr.) Also suppose the weight of the balloon, can, &c., is W lb, and let the barometer reading be h inches, then the ascensional power is equal to the weight of the air displace-weight of the air displaced -- weight of the heated air -- W lb, viz.,
FORMULA
080728 lb being the weight of a cubic foot of air at temperature 32°, under the pressure of one atmosphere, viz., when the reading of the barometer is 29.922 in. Of course, the motion depends upon the temperature of the air in the balloon as due to the furnace, if the latter is taken up with the balloon; but if the air in the balloon is merely warmed, and the balloon then set by itself, the problem is an easy one, as the rate of cooling can be determined approximately; but it is destitute of interest. We have said that dry air increase its volume by FORMULA part for every increase of 1o (Fahr.), but the air is generally more or less saturated with moisture. This second atmosphere, formed of the vapour of water, is superposed over that of the air, as it were, and, in a very careful consideration of the question, should be taken into account. Even, however, when the air is completely saturated with moisture but little difference is produced; so that for all practical purposes the presence of the vapour of water in the air may be ignored. Of course the amount of vapour depends on the dew-points, and tables of he pressure of the vapour of water at different temperatures are given in most modern works on heat; but, as has been stated, the matter, in an aeronautical point of view, is of very little importance. At first it was supposed that the cause of the ascent of the balloon of the Montgolfiers was traceable to the generation of gas and smoke from the damp straw which was set light to; but the advance of science showed that the fire-balloon owed its levity merely to the rearefaction of the air produced by the heat generated.
A formula giving the height, in terms of the readings of the barometer and thermometer, on the surface of the earth, and at the place the height of which is required, is easily obtained from the principle of hydrostatics. The formula given by Laplace, reduced to English units, is
FORMULA
Z being the height required in feet, h, h' the heights of the barometer in inches at the lower and upper stations, t, t' the temperatures (Farh.) of the air at the lower and upper stations, L the latitude, z the approximate altitude, and 20,886,900 the earth's mean radius in feet. This was the formula used by Mr. Glaisher for the reduction of his observations. It is open to the obvious defect that the temperature is assumed uniform, and equal to the mean of the temperature as the upper and lower stations; but till the law of decline of temperature is better determined, perhaps this is as good an approximation to the truth as we can have without introducing needles complication in the formula.
A sphere is not a developable surface -- i.e., it cannot be divided in any manner to as to admit of its being spread out flat upon a plane, so that no spherical balloon could be made of stiff plane material. However, the silk or cotton of which balloons are manufactured is sufficiently flexible to prevent any deviation from the sphere being noticeable. Balloons are made in gores, a gore being what, in spherical trigometry, is called a lune, viz., the surface enclosed between two meridians. The approximate shape of these gores is very easy to calculate.
Thus, let A B E C be a gore, then the sides A B E, A C E, are not arcs of circles, but curves of sines, viz., PQ bears to D B the ratio that sin A P does to sin A D, or, which comes to the same thing, supposing A D = 90°, and A P = xo, then P Q= B D sin xo. It is thus easy, by means of a table of natural sines, to form a pattern gore, whatever the required number of gores may be. Thus, supposing there are to be n gores, then B C must be a FORMULA of A E; and B D and A D being given, any number of points can be found on the curve A B E in the manner indicated above. A slight knowledge of spherical trigonometry shows the reason for the neck which is made to slope down, so that the whole shape resembles rather that of a pear. The pattern gore should originally be made as if for a spherical balloon, and afterwards the slight modification necessary for the formation of the neck should be applied.
The gores are sewn together, and a small portion of the upper end of each is cut away, so as to leave an aperture at the top of the balloon of from 1 to 3 feet in diameter. This space is occupied by the valve, which is generally made of strong wood, and consists of two semicircular shutters hinged to a diameter of the circular frame, and kept closed by a spring. The valve is opened by pulling a string, technically called the valve-line, which passes down through the balloon and out of the lower orifice in which the neck terminates. The net-work which, like the gores, is attached to the circumstances of the valve, passes over the surface of the balloon, and supports the ring or hoop from which the car is suspended by half a dozen strong ropes, of perhaps 4 or 5 feet in length. The network is thus stretched between the valve and the ring. It is very important that all the ropes by which the car hangs from the ring should be so adjusted that each may bear pretty nearly the same weight, as otherwise the whole netting and balloon will be strained, and perhaps to a serious extent. The car is usually merely a large basket made of wicker-work. The neck of the balloon should be 7 or 8 feet above the car, so that the aeronaut can easily reach it by mounting into the ring. The best material for the envelope is silk; but on account of the expense cotton or alpaca is generally used: in all cases it must be varnished, in order to render it more impervious to the gas. The graphel or anchor is a large five-pronged hook attached to the ring by a rope 100 or 120 feet long. The first care of the aeronaut on leaving the earth is to lower the graphel gently to the full extent that the rope will permit. Thus, when the balloon is in the air, the graphel hangs down below it and when the descent is being effected is the first thing to touch the ground. If the descent is well managed, and the balloon is moving downwards slowly, the weight of which it is relieved when the graphel is supported by the earth checks any further descent, and the wind carries the balloon along horizontally, the graphel trailing over the ground until it catches in some obstruction and is held fast. The balloon is then in much about the same position as a kite held by a string, and if the wind be strong, plunges about wildly, striking the ground and rebounding, until the aeronaut, by continued use of the valve-line, has allowed sufficient gas to escape to deprive it of all buoyancy and prevent its rising again.
The chief danger attending ballooning lies in the descent; for if a strong wind be blowing, the grapnel will sometimes trail for miles over the ground at the rate of ten or twenty miles an hour, catching now and then in hedges, ditches, roots of trees, &c., and, after giving the balloon a terrible jerk, breaking loose again, till at length some obstruction, such as the wooded bank of a stream, affords a firm hold. If the balloon has lost all its buoyant power by the escape of the gas, the car also drags over the ground. But even a very rough descent is usually not productive of any very serious consequences; as although the occupants of the car generally receive many bruises, and are perhaps cut by the ropes, it rarely happens that anything worse occurs. On a day when the wind is light (supposing that there is no want of ballast) nothing can be easier than the descent, and the aeronaut can decide several miles off on the field in which he will alight. It is very important to have a good supply of ballast, so as to be able to check the rapidity of the descent, as in passing downwards through a wet cloud the weight of the balloon is enormously increased by the water deposited on it; and if there is no ballast to throw out to compensate this accession of mass, the velocity is sometimes very great. It is also convenient, if the district upon which the balloon is descending appear unsuitable for landing, to be able to rise again. The ballast consists of fine baked sand, which becomes so scattered as to be inappreciable before it has fallen far below the balloon. It is taken up in bags containing about ¸ cwt. Each. The balloon at starting is liberated by a spring catch which the aeronaut releases, and the ballast should be so adjusted that there is nearly equilibrium before leaving, else the rapidity of ascent is too great, and has to be checked by parting with gas. It is almost impossible to liberate the balloon in such a way as to avoid giving it a rotary motion about a vertical axis, which continues during the whole time it is in the air. This rotation makes it difficult for those in the car to discover in what direction they are moving; and it is only by looking down along the rope to which the grapnel is suspended that the motion of the balloon over the country below can be traced. We may mention that the upward and downward motion at any instant is at once known by merely dropping over the side of the car a small piece of paper: if the paper ascends or remains on the same level or stationery, the balloon is descending; while, if it descends, the balloon is ascending. This test is so delicate that it sometimes showed the motion at a particular instant with more precision than did Mr. Glaisher's very delicate instruments.
Contrivances are often proposed by which the valve might be opened in less crude ways than by merely pulling a string attached to it; by which the jerks produced by the catching of the grapnel might be diminished, &c. These improvements are not adopted, because simplicity is requisite before everything. Any mechanical contrivance might be broken and rendered useless by the first blow of the car on the earth; whereas the primitive arrangements in use are such that scarcely any rough treatment can impair their efficiency.
The most important works that have appeared on the subject of aerostation are --
Dadaelus, or Mechanical Motions, by Bishop Wilkins, London, 1648; a Treatise on the Nature and Properties of Air and other Permanently Elastic Fluids, by Tiberius Cavallo, Lond 1781; Account of the First Aerial Voyage in England, in a Series of Letters to his Guardian, by Vincent Lunardi, London, 1784; History and Practice of Aerostation, by Tiberius Cavallo, London, 1785; Annals of some Remarkable Aerial and Alpine Voyages, including those of the author, by T. Forster, London, 1832; Aetronautica, by Monck Mason, London, 1838; A System of Aeronautics, comprehending its Earliest Investigations, by John Wise, Philadelphia, 1850; Astra Castra, Experiments and Adventures in the Atmosphere, by Hattou Turnor, London 1865; Voyages Aeriéns, par J. Glaisher, C. Flammation, W de Fonvielle, et G. Tissandier, Paris, 1870; the same translated into English and published, edited by James Glaisher, under the title, Travels in the Air, London, 1871.
All the above books we have seen ourselves, and used in the preparation of the present article. Astra Castra is a work of 530 pp. large quarto; it consists chiefly of extract from other works and writings, and it is useful as affording data for a history rather than as a history itself. On pp. 463-465 is a list of books and papers on aeronatics, which seems fairly complete up to the date 1864. In the list are also included memoirs and papers which we have not noted in the last paragraph, as the most important of them are referred to under their special subjects in the course of this article. We should advise any one desirous of studying the history of aeronautics to consult Mr. Turnor's list in Astra Castra, which is the most perfect we have met with. He has marked with an asterisk those works that may be consulted by the public in the library of the Patent Office, which contains, besides books, a valuable collection of prints and broadsheets on the subject of aerostation. (J. G.)
Read the rest of this article:
Aeronautics (Part 1) • Aeronautics (Part 2)
Author Information
The above article was written by James Glaisher, F.R.S., F.R.A.S., author of Travels in the Air.