f^ariidion uj'lhe Temperature. 397 



value of A :, the iacrement of the density, which is actually only 

 I 1 .^^^^^ Az _ dz _ 1 __ J 27300 27300 



27300 494/ Ay dy ^ 494/' 494/ 



= 1 - i, 494/= 27300: ( 1 -Pj , f= 55.26: (-^) 



= 55.26 f.. ^ J = 248, at the surface of the earth ; and since 



r = liand ^ == ZJL, wo have ^ = ZlLlI^ -537.6m 

 ar dr ps dz v 



whence, if we neglect the higher thixions of ^, we have for the point 

 at which the density is reduced to one half, putting Az = — i, 

 A^=268.Hp = .0759 ; and ^ becomes 1.3614, giving/= 208 

 feet, for the elevation corresponding to a variation of tempera- 

 ture of 1° at the supposed altitude, which would probably be 

 three or four miles above the surface. But, in order to com- 

 pute the height from the density, or from the pressure, with 

 greater convenience, it will be easiest to make the fluxion of y 

 constant, so that we may obtain a formula for the height as 

 depending upon the indication of the barometer. Thus, since 

 (ij. _ _ tl// dx _ _ 1 J dx _ dz __ dy , 



771Z dy mz dy mz- mz'-t, 



d-x 1 , .d'^x 2ztdz-\rz-dt 



- — = , whence d = — — 5 : ? =r 



dy" viz'-t, dy'^ mz*t,'- 



- ^^dy + z-tdr ^^^ d^ ^ -_2 T clr _ 



?)iz^ ^- dy-^ m^- ml^' dy 



p s y, 2 *i A A w , A V* 

 J. ^ — — : consequently Aa; =: — — 2- + — i [. 



m^^ V 711^- 7)1 2»t^ 



(--^ .- . ^' — ) Ay-', X = 1 being the radius of the 

 6/11^' V 3?« fv 



.1 • r * A a; A?/ , A w" A?/' 



earUi ; or.m leet, — = — — i + ^ — — ^ , 



20 900 000 766 1970 40371 



and Ax = —27300 Ay + 10608 Ay' - 518 Ay''; or, if D 

 be the descent of the barometer divided by the greater height, 

 //= 97300 D + 10608 D" + 518 D^, since Ay is negative, 

 and its square positive. Taking, for example, Z> = ^, we have 

 // = 13650 + 2652 + 05 = 16367 feet, or 3.1 miles, which 

 appears to be not far from the truth : at Quito the barometer 

 stands al 21.37, which gives a height of 8740 feet, instead of 



