FOR HIGH LFA'ELS IN THE EARTH's ATMOSPHERE. 35 



face to the next higlier one. Now tliis total quantity of work required is equal to 



X"^"' 



where by dz we designate an element of the vertical line from the point to sealevel and 

 by g designate the accleration of gravity for this element. We thus obtain the follow- 

 ing relation between V, g, and z : 



V=£ff-dz (1) 



where the integration is to be carried out along the vertical line joining the point with 

 sealevel. The distribution of gravity along the vertical and above the surface of the 

 earth is given by the well known formula 



g = </,(! - 0.000 000 095 7{z - sj), (2) 



wliere Zg represents the elevation of the earth's surface above sealevel, and g^ is the 

 acceleration of gravity at the earth's surface. If z represents depth below the earth's 

 surface then g at this depth is given by the formula 



ff = f/o(^ + 0.000 000 059 7(2„ - z)). (3) 



Here and in what follows, by the earth's surface in the neighborhood of a meteoro- 

 logical station is always meant the level of the barometer of the station, or the level 

 from which cloud-altitudes, kite-altitudes and the like are calculated [i. c, tlie so-called 

 "station level" of the United States Weather Bureau]. 



The ordinal number Vo of the gravity surface which coincides with the surface of 

 the earth at the station is obtained by substituting equation (3) in equation (1) and 

 integrating from sealevel up to the surface of the earth. We thus find 



T; = 0.464 876 x %x 2„(1 + 0.000 000 029 802,). (4) 



For example, to find V„ for the kite-station at Omaha, Nebr., we substitute the 

 altitude aljove sealevel, ^o = 1 241 feet, and the acceleration of gravity at the eartli's 

 surface at Omaha, g^ = 32.160 foot/sec.-, in formula (4) ; whence we have 



F„ = 0.464 876 x 32.160 x 1 241(1 + 0. 000 000 029 85 x 1 241) 

 = 18 550. 



There are thus seen to be 18 550 level surfaces of gravity between sealevel and the 

 level of the barometer of the kite-station at Omaha ; or work to the amount of 



pound X mile* 

 18 550*^ 



hour^ 



