77" 



MECHANICAL PHILOSOPHY. PNEUMATICS. 



[DENSITY or AIH. 



while the pressure on the condensed volume, B'C, U 

 that of a column of mercury of the tame base and of the 

 height 11 + K O. Now, wherever B' may be, it U 

 found by actu.U measurement that 



H +EO : II : : BO 1VO; 



and the pressures and volume* being as those linear 

 dimensions, it follows that 



Pressure on B' C : Pressure on B : : Volume B : 

 me B' C. 



And the same proportion is found to hold whatever be 

 the original density of the air experimented upon ; and 

 it equally has place for all elastic fluids. 



The foregoing law, although generally called the law 

 of Marritt, is equally entitled to be called Boyle's law ; 

 as it was announced, independently, by the latter philo- 

 sopher, at nearly the same time. It was discovered by 

 Iloylo in the year 1662. 



oe the force of compression on the air near the sur- 

 face of the earth is that due to the weight of the super- 

 incumbent atmosphere, it follows that the air must bo 

 rarer the higher we ascend ; rarer, for instance, at the 

 top of a mountain than at its base a truth that has been 

 turned to practical account in the way of measuring the 

 height of a mountain by observing, with the barometer, 

 the difference of atmospheric pressure at top and bottom. 



With the general theoretical principles and the practi- 

 cal construction of this useful instrument, we hero pre- 

 sume the reader to be already acquainted ; for although 

 some account of the theory of the barometer might 

 reasonably be expected in a treatise on Pneumatics, yet, 

 as the subject appertains chiefly to METEOROLOGY, and as 

 the space now at our disposal is limited, we must refer 

 for the requisite information, upon what concerns the 

 barometer aud thermometer, to that section. 



DENSITY OF THE ATMOSPHERE AT DIF- 

 KKKKNT HEIGHTS. Since the density of air varies 

 directly as the pressure sustained by it, the higher 

 regions of the atmosphere having a less superincum- 

 bent load than the lower, must be proportionally less 

 dense. This weight or pressure, as already noticed, will 

 be affected by temperature : aud moreover, the same 



particle, or volume of air, will weigh less, the high 

 is situated above the surface of the earth, on account uf 

 the diminution in the force of gravity. But, leaving 

 these comparatively unimportant modifying influence* 

 out of consideration unimportant, at leasi, for moderate 

 altitudes and regarding both the temperature and the 

 force of gravity as uniform, we may prove that The 

 density of the atmosphere at different heights above the 

 surface of the earth, varies thus namely, when the 

 heights increase in arithmetical progression, the densities 

 decrease in geometrical progression. 



Conceive a uniform slender column of the atmosphere 

 to be divided by horizontal sections into n equal parts ; 

 the number being so great, that each thin stratum of 

 air may be regarded as of uniform density : then com- 

 mencing at the bottom, or first stratum a,, the several 

 strata may be denoted by 



and their densities by 



(1) 



(2) 



It has been sufficiently explained (HYDROSTATICS, page 

 750) that, in speaking of the density of any substance, 

 we always have reference to some assumed unit of 

 density, and that the weight of the unit of volume (the 

 unit of weight) of the standard substance, multiplied by 

 the numerical expression (D) for the density of any other 

 substance, gives the weight of a unit of volume of that 

 other substance. The abstract numbers (2) therefore 

 will equally express the number of units of weight of 

 the volumes (1), provided we take the magnitude of 

 Oj or Ojj, &c. , for that of the unit of volume : and this 

 we are of course at liberty to do, for in all investigations 

 our unit of measure may be chosen at our convenience. 



It may be well, however, in order to prevent confusion, 

 to write the numbers (2), when regarded as so mauy 

 units of weight, thus, namely : 



Then, as the density of any stratum of air varies with 

 the weight or pressure it sustains, we have 



r 



' ' 

 ''4 



But 



+ 



-. -. 4- 

 ...+ 

 <bc. 



) 



I! 1 



-f 1C 4 + + W: 



4- u), 4- ... 4- w, 

 "'"Ac. 



(3). 



<*s 4 : w s >4 ; 



Hence, putting to for d, in the foregoing proportions, alternating and compounding the ratios (ALGEBRA, page 

 477), we have 



+ w, + w, 4- + - - - 4- 



>g + w, + tc 4 + 



+ . . . 4- 

 4- -j- 



u> 



3 4- 

 >4 4- 



4- 



Consequently, alternating, and having regard to the 

 proportions (3) and (4), we have 



d, : dj | : : d, : rf 8 : : d s : d 4 , Ac. ; 

 therefore the densities of the strata a,, a,, a,, a t , <fea, 

 of which the heights are in arithmetical progression, are 

 themselves in geometrical progression ; and consequently 

 the pressures sustained at different altitudes are also in 

 geometrical progression. 



I (NATION OF ALTITUDES BY THE 

 BAROMETER. Let A be the altitude in feet of any 

 pot above the surface of the sea ; then measuring down- 

 wards from that spot, the distances 0, 1, 2, 3, Ac., feet, 

 the preHUTM upon each foot will form a geometrical 

 Increasing progression, by what is proved above. The 

 relation, therefore, between the numbers expressing this 

 progiession, and the numbers 0, 1, 2, 3, Ac. , U the same 

 as that between any numbers in geometrical progression, 

 and their logarithms. The numbers 0, 1, 2, 3, express- 

 ing the distances downwards from the highest point, 

 must therefore be the common logarithms of the num- 

 bers expressing the corresponding pressures, multiplied 

 by anrao constant factor or modulus, which constant 

 factor we may call K. 



Hence, a expressing any lower altitude in fei-t, and 

 P, p the pressures at the altitudes A, a, we shall have 



A o K log. P log. p = K log. 



If the lower station be at the level of the sea, thru 

 a ; and since the pressures P, /> are indicated by the 

 heights of the mercurial column in the barometer at the 

 two stations, we have for A, the number of feet in the 

 altitude of the upper station above the lower, 



A-Klog. --Klog 

 P 



But before this formula can be turned to pra. 

 account, wo must be able to assign the numerical value 

 of the constant multiplier K. 



Let H be the height of the barometer at the level of 

 the sea, and H' the height at some known altitude A'; 

 then 



A' - K log. g> - K log. 5. 



Let A' - 1 foot, then 



