It has been found that the volume of gas disengaged from gun powder is equal to about 300 times the volume of the powder itself ; and that its expansive force, when increased by the heat which is generated at the time of the explosion, is about 1500 times as great as the pressure of the atmosphere in its ordinary state. It must con sequently exert a pressure againsta cannonball, and the interior of the chamber of the gun, equal to about 15,000 pounds upon every square inch of the surface upon which it acts.
The fact that the density of air varies with the compressing force is sufficient to show that the atmosphere about the earth cannot he of uniform density ; and it is also evident that the density must dimiuish from the surface of the earth upwards, according to some law depend ing on the height of any point above the earth, or rather upon the weight of the mass of air above that point. It might, at first, be supposed that the atmosphere would extend upwards to a height at which the centrifugal force of the particles of air (by the diurnal revolution) is equal to the force of gravitation ; and it is shown by Poisson (` Tmit4 de .116canique," tom. ii., 619) that, conformably to this principle, the height of the atmosphere at the equator should be equal to about fivo times the semi-diameter of the earth. But it is probable that, long before this height is attained, the air loses its elasticity by the cold in the upper regions, or that its expansion La destroyed by the pressure of the ethereal fluid which is diffused through infinite space. By the duration of twilight it is inferred that the atmosphere is capable of reflecting the sun's rays at the height of about 45 miles above the earth, and it is probable that some light is reflected from a still more elevated region.
In order to determine the law by which the density of the atmo sphere diminishes at increasing distances from the earth's surface, on the supposition that the action of gravity and the temperature of the air are constant, let r be the centre of the earth, and let A z La the height of a very slender column of air extending vertically upwards to tho top of the atmosphere. Also let the atmosphere be divided into an infinite number of concentric: strata of equal thicknesses, which latter are represented by A a, B C, C D, Se.; and, as these thicknesses are small, let the of the air in each stratum be supposed uniform and equal to that which is due to the weight of all the strata above it.
Let d„ &c., represent the densities of the several strata whose heights are eB, B c, c D, &c.; these terms may also represent the weights of the slender columns A B, B C, C D, &C. ; consequently the weights of the columns A B, A C, A D, &c., may be respectively repre sented by d„ d, + d, + d„ &c. Then, the density in each stra tum being proportional to the weight, or sum of the densities, of all above it, we have and so on.
Thus d„ d,, &c., are in a geometrical progression decreasing.
Now A u, A C, A D, &c., form an arithmetical progression increasing ; or, reckoning both the heights and the densities from any point, as x, downwards, the former (that is, K H, K G, K &c.) form an arithmetical progression, and the densities in x )1, n o, GP, &c., form a geometrical progression; both increasing. But a series of numbers in an arith metical progression being made to correspond to a series in geometrical progression, the former numbers are logarithms of the latter; and thus the distances K x G, K F, &c., may be considered as representing the
logarithms of the densities in the strata K B, n 0, 0 F, &C., respectively.
Imagine any point K to be the origin of the abscissa) (represented by x) on the vertical line z w ; and imagine any horizontal ordinates .x Ff, by y) to be drawn; then, if K F, K D, &c., be pro portiona to the logarithms of rf, ad, fte., the lino a dfk, &c., is called the logarithmic curve, and its equation is log. y= x log. a, or y =a. [Loaeatrimic Cuays), where a is some constant which is called the base of the system of logarithms appertaining to the par ticular curve.
Now it has been demonstrated by mathematicians, that if tangents am, d n, &c., be drawn from any points in the curve, the subtangents m, m n, &c., will be ecpial to one another ; and that the area corn• prehended between the infinite branch a z of the curve, its asymptote A z, and any ordinate A a, D d, &e., is equal to the product of the con stant subtangent, or modulus of the curve, and that ordinate : hence the area between • a and the infinitely remote summit z is equal to ease m. Also, by the nature of logarithms, the logarithms of the same natural number in different systems of logarithms bear to one another the same proportion as the moduli of those systems. We have therefore only to find the value of the subtangent w m, or modulus, for what may be -called the atmospherical logarithms. For this pur pose, let h denote the height of a homogeneous atmosphere whose density is equal to that of the real atmosphere at the surface of the earth, which density is represented by the line A a in the above dia gram ; then hx e a will represent the weight of such homogeneous atmosphere, or its pressure on the point A. But the area between z e, a, and the curve being supposed to be made up of the infinite number of ordinates A a, D d, F f, &c., which, severally, represent the densities of the air at the points A, D, F, &e., in the infinitely high column A Z of atmospherical air ; that area, namely, wax A 1/1, may represent the weight of such column, or the pressure of the real atmosphere on the point. A ; this being made equal to the former pressure, it is evident that we shall have e m= h. Thus the height of a homogeneous atmosphere exercising at w the same pressure as the real atmosphere, will he the subtangent, or modulus, of the atmo spheric logarithms. The value of it is determined by a proportion in which the heights of the column of homogeneous air, and the column of mercury which holds it in equilibrio, are to ono another inversely as the specific gravities of the two fluids. [Hvariosreves.] Now the specific gravities of air and mercury being, respectively, proportional to about and 18568 ; and the height of the column of mercury in the barometer being 30 inches when the temperature is expressed by Fehr., we get 27803 feet, for the value of h. Hence the height of a homogeneous atmosphere would be about 27803 feet, or a little more than 5 miles from the sea-level, while the mean density of the atmosphere (in its present state) is at about 3˘ miles.