PNEUMATICS. 



PNEUMATICS. 



590 



Let d v d t , dj, &c., represent the densities of the several strata whose 

 heights are AB, BC, CD, &c. ; these terms may also represent the 

 weights of the slender columns A B, B c, CD, &c. ; consequently the 

 weights of the columns A B, AC, AD, &c., may be respectively repre- 

 sented by d,, rf, + d 2 , rf, + rf 2 + d 3 , &c. Then, the density in each stra- 

 tum being proportional to the weight, or sum of the densities, of all 

 above it, we have 



: d s : rf 3 + d t + d 5 + &c. (I) 



In like manner 



. : : d, : d, + d 5 



or, by composition, 



d, : d, + d, + d t + d, + kc. : : 



In like manner 



::d t : d t + d, + d, 



(II) 



d i + &c. (Ill) 



(IV.) 



Then, from (I) and (III), by equality of ratios, we have 



d, : d, : : d, : d,. 

 And, from (II) and (IV), by equality of ratios, we have 



d, :d 3 ::d 3 :d t ; 

 and so on. 

 Thus d,, d,, dj, d t , Ac., are in a geometrical progression decreasing. 



Now A B, A c, A D, &c., form an arithmetical progression increasing ; 

 or, reckoning both the heights and the densities from any point, as K, 

 downwards, the former (that is, KH, KG, KF, &c.) form an arithmetical 

 progression, and the densities in KH, Ho, GF, &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 H, K G, K F, tc., may be considered as representing the 

 logarithms of the densities in (he strata K H, H o, OF, &c., respectively. 



Imagine any point K to be the origin of the abscisse (represented by 

 x) on the vertical line z A ; and imagine any horizontal onlinatcs K k-, 

 F/, &c. (represented by y) to be drawn ; then, if K F, K D, &c., be pro- 

 portional to the logarithms of F/, Dd, &c., the line adfk, tc., is 

 called the logarithmic curve, and its equation is log. y=.f log. a, or 

 y = a* [LOGARITHMIC CTRVK], 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 gubtangenta 

 Am, mn, &c., will be equal to one another; and that the area com- 

 prehended between the infinite branch a z of the curve, its asymptote 

 A z, and any ordinate A o, D d, tic., is equal to the product of the con- 

 stant subtangent, or modulus of the curve, and that ordinate : hence 

 the area between A a and the infinitely remote summit z is equal to 

 A a x A 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 A m, or modulus, 

 for what may be -called the atmospherical logarithms. For this pur- 

 pose, let A denote the height of a homogeneous atmosphere whose 

 density ia equal to that of the real atmosphere at the surface of the 

 earth, which density ia represented by the line A a in the above dia- 

 gram ; then A x A a will represent the weight of such homogeneous 

 atmosphere, or its pressure on the point A. But the area between z A, 

 A a, and the curve being supposed to be made up of the infinite 

 number of ordinates \a, Dd, F/, &c., which, severally, represent the 

 densities of the air at the points A, D, F, ftc., in the infinitely high 

 column A z of atmospherical air ; that area, namely, A a x A m, 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 Am = A. Thus the height of 

 a homogeneous atmosphere exercising At A the same pressure as the 

 real atmosphere, will be the subtangent, or modulus, of the atmo- 

 spheric logarithms. The value of A 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 one another inversely 

 as the specific gravities of the two fluids. [HYDROSTATICS.] Now the 

 specific gravities of air and mercury being, respectively, proportional 

 to about 1'22 and 18568 ; and the height of the column of mercury in 

 the barometer being 30 inches when the temperature is expressed by 

 55 Fahr., we get 27803 feet, for the value of k. 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!, i 



The mountain-barometer, as it is called, is usually provided with an 

 adjusting screw, by which the surface of the mercury in the cistern 

 may be made to coincide with the zero of the scale of inches by which 

 the height of the column is expressed ; but those of a more portable 

 kind have not that adjusting screw, and then a correction must be 

 made for the error of the scale. [BAROMETER.] 



Water boils when the elastic power of the vapour formed from it is 

 equal to the incumbent pressure ; and consequently the temperature 



;\ 



- ) given by Mr. Tredgold, might be employed. Here 



t is 



at which the boiling takes place in the open air will depend upon the 

 weight of the atmospheric column above it. Therefore, since this 

 weight becomes less as the station is more elevated, it is evident that 

 water will boil at a lower temperature on a mountain than on the 

 plain at its foot ; and the Rev. Mr. Wollaston constructed an instru- 

 ment called a thermometrical barometer, by which, on the principle 

 just mentioned, the relative heights of stations can be found. A tube 

 containing the mercury is provided with a graduated scale, and, when 

 used, the bulb is placed in a vessel of water, which is made to boil by 

 means of a spirit-lamp. An improved form of the apparatus is repre- 

 sented under BOILING OF LIQUIDS. 



In order to determine the heights of stations merely by the know- 

 ledge of the temperature at which water boils, the formula P = 

 /t+75' 

 \ 85 



the temperature of the boiling water at the station, expressed in 

 degrees of the centigrade thermometer ; r is the measure of the elastic 

 force of the steam at the temperature ( under the pressure of the 

 atmosphere, and is expressed by the corresponding height, in centi- 

 metres, of the column of mercury in a barometer. 



The velocity with which air flows into a vacuum through an aperture 

 in a vessel follows the same law as water or any other non-elastic fluid 

 [HYDRODYNAMICS] ; for though, in the former case, the quantity of air 

 passing through the orifice in a given time varies with the density of 

 that which successively comes to the orifice, yet the pressure by which 

 the air is forced out varying in the same proportion, the velocity, by 

 dynamics, remains constant. 



Now, the velocity acquired by a falling body in vacuo is known to 

 be=V2gx, if g= 32 feet, and x = height fallen through. (DYNAMICS.) 

 Hence, if v = velocity with which the air rushes through a small hole 

 into a vacuum at the sea-level, v = V-gx, namely, v = the velocity 

 of a particle of air supposed to have fallen from the height of a 

 homogeneous atmosphere into the aforesaid hole. So that v= 

 'J x 27803, from above. 



. . v = about 1339 feet per second. 



The law is the same, whether we consider the air to act only by its 

 weight, or whether it be confined in a vessel and the efflux be produced 

 by the elasticity. For, the air in the vessel being in the ordinary state 

 of the atmosphere, the pressure against every point on the interior 

 surface is equal to the pressure of the atmosphere by which, if not 

 otherwise confined, it would be kept in its actual state ; consequently 

 it begins to flow from the orifice with the same velocity as if it had 

 been impelled by the weight of the whole column of atmosphere above 

 the orifice, that is, with the velocity due to the descent of a body from 

 a height equal to that of a homogeneous atmosphere. After this 

 moment, the density of the air in the vessel diminishing, its elasticity 

 diminishes with it, and consequently the power of motion is diminished 

 in the same ratio as the density. It may be added also that, since 

 density of air increases with the pressure, an additional pressure on the 

 fluid in a vessel will not increase the velocity of the efflux. But the 

 law just mentioned only holds good when the vacuum is supposed 

 to remain perfect on the exterior of the orifice : for, if the air bo 

 received in a vessel, it will expand in that vessel and re-act against the 

 effluent air at the orifice, thus diminishing the velocity till the latter 

 finally becomes equal to zero ; and this will take place when the air 

 has attained the same density in the two vessels. 



If the effluent air be of a given density, but not the same as in the 

 ordinary state of the atmosphere, the force by which it would be made 

 to flow into a vacuum must be determined by the above equation 

 PD' = P'D ; where ! is the pressure (or weight of the column) of the 

 ordinary atmosphere, and D its density at the earth's surface ; D' is the 

 given density and r' is the required pressure or force by which that 

 air would be impelled through the orifice. Now if air in the ordinary 

 state be allowed to rush into a vessel containing air less dense than 

 itself, and the velocity of efflux be required, the moving force will be 

 the difference between that with which the ordinary air is driven 

 through the orifice and that with which the rarer air would be so 

 driven ; that is, it may be represented by p p' ; then the velocities 

 of efflux being as the square roots of the forces [HYDRODYNAMICS], if 

 the Velocity ilue to the force p is given, the required velocity at the 

 commencement of the efflux may be found. 



The determination of the velocity with which dry steam or any 

 other elastic fluid rushes into a vacuum, or into a fluid of less density 

 than itself, is made in the same manner as for air. Thus, knowing the 

 temperature of steam, and consequently its elasticity, or the equiva- 

 lent pressure, we can find the height of a homogeneous atmosphere 

 which would produce the game pressure ; and then the velocity with 

 which the steam flows into a vacuum would be equal to that acquired 

 by a body in falling down the height of such atmosphere. But if the 

 steam is to flow into any elastic fluid of less density than itself, the 

 height of the homogeneous atmosphere must correspond to the 

 difference of the pressures arising from the different elasticities of the 

 two fluids. 



Finally, it has been lately found by some very accurate experiments 

 recorded by Professor Potter, that, if we represent by i> the theoretic 



