HEIGHTS DETERMINED BY THE BAROMETER. 125 



the foot to the top of a mountain, and will rise again when 

 it is returned to its former position. 



(1) When the force of gravity is regarded as con- 

 stant. 



Consider a vertical column of the atmosphere at rest 

 under the action of gravity. Let z be taken vertical and 

 positive upwards ; and at a height z, let p be the pressure 

 and p the density. The pressure p, at any height z, is meas- 

 ured by the weight of the column of air extending from 

 that height to the top of the atmosphere; and the element- 

 ary pressure dp will be measured by the weight of the col- 

 umn having the same base and the elementary height dz. 

 Therefore, if A be the area of the section of the column, we 

 have 



Adp = Agp dz, 



or, dp = gpdz, (1) 



the negative sign being taken because the pressure p is a 

 decreasing function of the height z. 



If t be the temperature, we have from (3) of Art. 54, 



p = tp(l+ at}. (2) 



Dividing (1) by (2), we have 



dp _ gdz 



* p ' ~l+t 



If the heights above the earth's surface are small, the 

 force of gravity g may be regarded as constant ; and sup- 

 posing t constant, we have, by integrating (3), 



k W P -9 V - *) m 



lo Sy- T+~5~' 



where p' is the pressure at the height z'. 



