BAR 



BAR 



metical progreffion, the oidlnates yJE, BF, CG, DH, will 

 be in geomtlrical piogi-cffion. It is another fundamental 

 property of this curve, that if EK or HS touch the 

 curve in E or H, the fubtangent ulK or DS is a conftant 

 quantity. Moreover, the infinitely extended area M/IEN 

 is equal to the redanglc KjIKL of the ordinate and fub- 

 tangent ; and the area MDHN is equal to SD X DH, 

 or to K/l X DH ; and, thertfore, the area lying beyond 

 any ordinate is proportional to that ordinate. Thefe pro- 

 perties are analogous to the principal circiimflances in the 

 conftitution of the atmofphere, on the fuppofition of equal 

 gravity. The area MCGN reprefeiUs t-ie whole quantity 

 of aerial matter above C, for CG is the denfity at C, and 

 CD is the tliickncfs of the ftratum between C and D, and, 

 therefore, CGHD will be as the q'.;sntity of air in it, and fo 

 of all the others, and of their fums, or of the whole area 

 MCGN ; and as each ordinate is proportional to the area 

 above it, fo each denlity, and the quantity of air in each 

 ftratum, is proportional to the quantity of air above it ; and 

 as the whole area MJEN is equal to the reftangle KAEL, 

 fo the whole air of variable denfity above A might be con- 

 tained in a column KA, if, inllead of being comprefled by 

 its own weight, it were without weight, and comprtfied by 

 an external force equ;d to the preflure of the air at the fur- 

 face of the earth ; and, in this cafe, its uniform denfity 

 would be cxprciTcd by AE, the meafure of the dei:rity at 

 the furface of the earth, and it would form what may be 

 called the homogeneous atraofphere. Hence it follows, 

 that the height of this atmofphere is the fubtangent of that 

 curve, whole crdinat'-s are as the deufities of the air at dif- 

 ferent heights, on the fuppofition of equal gravity. In or- 

 der to determine this fubtangent, we may con:pare the den- 

 fities of mercury and air ; for a column of air of uniform 

 denfity, reaching to the top of the homogeneous atmo- 

 fphere, counterbalances the mercury in the barometer. 

 From the bell experiments it is inferred, that when mercury 

 and air are of the temperature of 32" of Fahrenheit, and 

 the barometer ftands at 30 inches, the mercury is nearly 

 10440 times dcnfer than air; confcquently the height of 

 the homogeneous atmofphere is 10440 x 30 inches = 

 313200 inches = 261CO feet = 8700 yards =: 5 miles 

 wanting 100 yards. Or we may find this height by ob- 

 ferving the variations of the barometer at known altitudes, 

 thus ; when the mercury and air are of the above tempera- 

 ture, and the barometer on the fea-diore ftands at 30 inches, 

 an afcent of 883 feet will caufc it to fall to 29 inches. 

 Moreover, in all logarithmic luri'es having equal ordinates, 

 the portions of the axes intercepted betwten the correfpond- 

 ing pairs of ordinates, arc proportional to the fubtangents ; 

 and the lubtangent of the curve belonging to our common 

 tables is 0.4342945 ; and the difference of the logarithms of 

 30 and 29, which is the part of the axis intercepted between 

 the ordinates 30 and 29, or 0.0147233 : 0.4342945 :: 883 

 : 26046 feet =: 8680 yards ;= 5 miles wanting 120 yards, 

 diflering from the former refult 20 yards. This difference 

 refulls from the difficulty of accurately afcertaining the re- 

 fpeftive deniities of mercury and air, and alfo of duly elli- 

 mating the elevation which caufes a fail of one inch in the 

 barometer. This inveftigation, however, proceeds upon the 

 fuppofition of equal gravity ; whereas it is well known, that 

 the weight of a particle of air decreafes as the fquare of its 

 diftance from the centre of tiie earth increafes. In order, 

 therefore, that a luperior ftratum may produce an equal 

 preffure at the furface of the earth, it mull be denfcr, be- 

 caufe a fmgle particle of it gravitates lefs ; confequentlv, the 

 denfity at equal elevations muft be greater than on the fup- 

 pofition of equal gravity, and the law of its diminution muft 

 be diiTereiit. 



Make OD ; OA :•. OA '. Od% 



OC : OA :: OA : Or 



OB : OA :: OA : Oi, Sec : fo that 

 a/, Of, Oi, OA, may be reciprocals to OD, OC, OB, OA ; 

 and through the points A, b, c, il, draw the perpendiculars 

 AE, If, eg^, till, proportional to the deniities in A, B, C, D; 

 and let CD be fuppofed exceedingly finall, fo that the den. 

 fily may be fuppofed uniform through the whole ftratum. 

 Then we (hall have, OD X Od — OA' = OC x Oc ; and 

 Oc : Od :: OD : OC ; and Ot : Oc - OJ :: OD : OD-OC, 

 or Oc : cdi: OD : DC, and cd : CD :: Oc : OD; or be- 

 caufe OC and OD are ultimately in the ratio of equality, 

 we have cd : CD :: Oc : OC :: OA' : OC, and cd = CD 

 OA' _ . __ OA' 



^Uc?' ""^ '^'^ ^ ^■^ 



CD X eg X 



OC 



but CD X 



c-r X is as the preffure at C arifintjc from the abfolutc 



OC- 



weight of the ftratum CD ; for this weight is as the bulk, 



as the denfitv, and as the gravitation of each particle joint- 



OA' 

 ly. But CD expreftes the bulk, eg the denfity, and . — _ 



the gravitation of each particle. Confeqnently cd X eg is 

 as the prefi'ure on C arifing from the weigiit of the ftratum 

 DC ; but cd X eg is evidently the element of the curvilincal 

 area AmiiE formed by the curve Efghn, and the ovdi: atcs 

 AE, If, eg, ah, &c. 1)111. Therefore the fnm of all the ele- 

 ments fuch as cdhg, that is the area cmng below ^5-, will be 

 as t'le whole preffure on C, arifing from the gravitation of 

 all the air above it ; but by the nature of air, this whole 

 prefture is as the deniity which it produces, that is, as eg. 

 Hence it appears that the curve Egn is fuch, that the area 

 lying below or beyond any ordinate eg is proportional to 

 that ordinate; and this being the property of the logarithmic 

 curve, Egn is a curve of this nature. Befides, this curve is 

 the fame with EGN; for let B continually approach to A, 

 and ultimately coincide with it. It is evident that the ulti- 

 mate ratio of BA to Ab, and of BF to bf, is that of equa- 

 lity ; and if EFK, Efk, be drawn, they will contain equal 

 angles with the ordinate AE, and will cut off equal fubtan- 

 gents AK, Ak, The curves £GA'^, Egn, are, therefore, the 

 fame in oppofite pofitions. Moreover, if OA, Oh, Oc, Od, 

 Sec. be taken in arithmetical progreifion dccreafing, their 

 reciprocals OA, OB, OC, OD, &c., will be in harmonical 

 progreffion increafing (fee Progression) : but, from the 

 nature of the logarithmic curve, when OA, Ob, Oc, Od, &c. 

 are in arithmetical progreffion, the ordinates AE, bf, eg, db, 

 &c. are in geometrical progreflion. Confequently, when 

 OA, OB, OC, OD, &c. arc in harmonical progreifion, the 

 denfities of the air at A, B, C, D, &c. are in geometrical 

 progreflion; and thus the ilcnfitits of the air at all eleva- 

 tions may be difcovered. Thus, to find the denfity of the 

 air at K, the top of the homogenous atmofphere, make OK 

 : OA :: OA : OE, and draw the ordinate LT ; LT is the 

 denfity at K. 



The correftion for the diminidied gravity of the air ftated 

 by profcfTor Playfair (Edinb. Tranf. vol. i. p. 118.) is a 

 third proportional to the lemidiametcr of the earth, and 

 the height as computed by the ordinary rule ; and for dif- 

 ferent mountains, this correftion is in the duplicate ratio of 

 their heights. Dr. Horfley finds (Phil. Tranf. vol. Ixiv.), 

 that in a height of 4 Engliih miles, the diminution of den- 

 fity or volume from the aceelerative force of gravity would be 

 only -i^cTjth part of the whole, or about 48 feet ; and this af- 

 feft, being in the duplicate ratio of the heights, becomes at 

 one mile high only three feet. Below the furface of the 

 earth, it is but half the quantity ; gravity within the earth 

 being fimply as the diftance from the centre. 



As 



