Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/224

and ${\displaystyle {\text{C}}}$ is determined as before by putting ${\displaystyle x=0,}$ when we have ${\displaystyle u=u_{0}.}$

In this case ${\displaystyle u_{0}}$ 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 ${\displaystyle h,}$ at which it becomes quite full, when the motion changes, and is as just investigated. Then ${\displaystyle u_{0}}$ becomes the velocity at the height ${\displaystyle h,}$ and everything is measured from this height as if from the surface of the earth, ${\displaystyle a}$ being then the radius of the earth ${\displaystyle +h,\rho _{0},\sigma _{0}}$ the densities at height ${\displaystyle h,}$ and ${\displaystyle \rho ,\sigma }$ at height ${\displaystyle x+h,}$ &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.

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 ${\displaystyle {\tfrac {1}{273}}}$d part of its volume for every increase of temperature of ${\displaystyle 1^{\circ }}$ centigrade, or ${\displaystyle {\tfrac {1}{491}}}$th of its volume for every increase of temperature of ${\displaystyle 1^{\circ }}$ Fahr. If, therefore, the air in an envelope or bag be heated ${\displaystyle 60^{\circ }}$ Fahr. more than the surrounding air, the air within the bag will expand ${\displaystyle {\tfrac {60}{491}}}$th of its volume, and this air must therefore escape. The air within the bag weighs less, therefore, than the air it displaces by the ${\displaystyle {\tfrac {60}{551}}}$th part of the latter; and if the weight of this be 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 ${\displaystyle {\text{V}}}$ cubic feet of hot air at the temperature ${\displaystyle t^{\prime }}$ (Fahr.), and let the temperature of the surrounding air be ${\displaystyle t}$ (Fahr.) Also, suppose the weight of the balloon, car, &c., is ${\displaystyle {\text{W}}}$ ℔, and let the barometer reading be ${\displaystyle h}$ inches, then the ascensional power is equal to the weight of the air displaced ${\displaystyle -}$ weight of the heated air ${\displaystyle -{\text{W}}}$ ℔, viz.,

${\displaystyle .080728}$ ℔ being the weight of a cubic foot of air at temperature ${\displaystyle 32^{\circ },}$ under the pressure of one atmosphere, viz., when the reading of the barometer is ${\displaystyle 29.922}$ 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 free 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 increases its volume by ${\displaystyle {\tfrac {1}{491}}}$th part for every increase of ${\displaystyle 1^{\circ }}$ (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-point, and tables of the 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 rarefaction 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 principles of hydrostatics. The formula given by Laplace, reduced to English units, is—

${\displaystyle {\text{Z}}}$ being the height required in feet, ${\displaystyle h,}$ ${\displaystyle h^{\prime }}$ the heights of the barometer in inches at the lower and upper stations, ${\displaystyle t,}$ ${\displaystyle t^{\prime }}$ the temperatures (Fahr.) of the air at the lower and upper stations, ${\displaystyle {\text{L}}}$ the latitude, ${\displaystyle z}$ the approximate altitude, and ${\displaystyle 20,886,900}$ 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 temperatures at 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 needless complication in the formula.

A sphere is not a developable surface—i.e., it cannot be divided in any manner so 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 trigonometry, is called a lune, viz., the surface enclosed between two meridians. The approximate shape of these gores is very easy to calculate.

Thus, let ${\displaystyle {\text{ABEC}}}$ be a gore, then the sides ${\displaystyle {\text{ABE}},}$ ${\displaystyle {\text{ACE}},}$ are not arcs of circles, but curves of sines, viz., ${\displaystyle {\text{PQ}}}$ bears to ${\displaystyle {\text{DB}}}$ the ratio that ${\displaystyle \sin {\text{AP}}}$ does to ${\displaystyle \sin {\text{AD}},}$ or, which comes to the same thing, supposing ${\displaystyle {\text{AD}}=90^{\circ },}$ and ${\displaystyle {\text{AP}}=x^{\circ },}$ then ${\displaystyle {\text{PQ}}={\text{BD}}\sin {x^{\circ }}.}$ 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 ${\displaystyle n}$ gores, then ${\displaystyle {\text{BC}}}$ must be ${\displaystyle {\tfrac {1}{n}}}$th of the circumference—viz., ${\displaystyle {\tfrac {2}{n}}}$ths of ${\displaystyle {\text{AE}};}$ and ${\displaystyle {\text{BD}}}$ and ${\displaystyle {\text{AD}}}$ being given, any number of points can be found on the curve ${\displaystyle {\text{ABE}}}$ in the manner indicated above. A slight knowledge of spherical trigonometry shows the reason for the above rule. Balloons, as usually constructed, are spherical, except for the neck, which is made to slope down, so that the whole