42 CONSTRUCTION OF ISOBARIC CHARTS 



Such a map for North America, constructed by the aid of this table, is shown in 



PI. VIII. The curves of Vq on this map show that by reason of gravitation it always 



mile" 

 requires the performance of work amounting to 10 000 . ^ in order to raise the 



unit mass from a point on one curve to any point on the curve next above. 



III. The Relative Positions of the Isobaric Surfaces and the Level Sur- 

 faces OP Gravity under Static Conditions. 



The well-known condition for atmospheric equilibrium is that the isobaric sur- 

 faces and the level surfaces of gravity shall coincide. If this condition is fulfilled then 

 we may express the pressure p as a function of the gravity-potential only ; and con- 

 versely can write the gravity-potential V as a function of the pressure only. In the 

 following pages pressure considered as a function of gravity-potential will be repre- 

 sented by py, and gravity-potential considered as a function of pi'essure will be 

 represented by Vp. The values of these functions are obtained by integrating the 

 differential equation for the barometric determination of heights.* Since it is conve- 

 nient to perform these integrations at first for special intervals, the following expres- 

 sions are introduced : 



e::=t;,-t;„ (7) 



^ll = Pv.-Pn- (8) 



According to the above given definitions the quantities Vp^ and Vp^ are equal to the 



mile^ 

 number of level surfaces of gravity expressed in ■, - -o units lying between sealevel and 



the isobaric surfaces p^ and 2h respectively ; and E^^ is the number of level surfaces 

 between the two isobaric surfaces po and pi. The quantities p^^ and p^^ are the num- 

 bers of isobaric surfaces lying between sealevel and the two level surfaces of gravity 

 numbered Vq and Vi respectively. n|'' represents the number of isobaric surfaces 

 lying between the two level surfaces of gravity Vq and Vi. In all this we imagine the 

 existence in the atmosphere of an isobaric surface for each inch of the column of a 

 mercurial barometer [under standard gravity]. 



To calculate E^J we start with the equation of condition for dry air, viz.: 



f = ^f» (9) 



-* -'o 



and with the differential equation for the barometric measurement of altitudes, viz.: 



* Note by the Editor : All barometric readings and isobars refer to absolute pressures as indicated by the mer- 

 curial column reduced to standard temperature, gravity, etc. 



