Height of the Balloon Determined by Readings of the Barometer and Thermometer
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
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.
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