374 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



Therefore, if we have given a system of surfaces of constant pres- 

 sure corresponding to the pressures ft; ft + A ft; ft + 2 J/?, etc., then 

 the vertical distance between the neighboring surfaces at any point 

 is inversely proportional to the prevailing density of the air at this 

 point. 



This latter theorem corresponds to that for the isobars according 

 to which the gradient is obtained from the reciprocal of the distance 

 between neighboring isobars. 



If now for p we substitute in the above expression 



„ p T « 



P " ft T 



then equation (3) becomes 



13.6 ft T 

 Ah-— -.7? tT Aft. 



Po P ] 



Hence it follows that 



"The vertical distance between two definite isobaric surfaces/? and 

 ft + A ft at any point is proportional to the absolute temperature 

 prevailing at that point." 



Finally, if in equation (ic) in place of the expression 



13.6 it 



P 



we substitute the difference A h from equation (3) we obtain 



Ah 



r = T g - 



Now since I is the distance between the pressure surfaces /? and 

 {3 + A ft measured horizontally from a point on the surface ft, 

 whereas Ah is the distance of the same surfaces measured vertically, 



Ah. 



therefore — is the tangent of the angle of inclination of the surfaces 



or tg a. From this consideration we obtain the well-known equa- 

 tion (2) above given or / = g tg a. This equation is distinguished 

 by its simplicity from all of those to which the number (1) was given 

 in the preceding paragraph. 



Translated into words this equation (2) would read 

 "The gradient acceleration is proportional to the inclination of the 

 isobaric surfaces." 



