ULTRA-SHORT-WAVE TRANSMISSION PHENOMENA 387 



and the percentage of water vapor by 



a = - OAOSh + 1.372. 



The water vapor percentage gradient increases (in absolute value) up 

 to 1.25 kilometers, after which it decreases; the other two curves 

 (" r " and " p ") do not have a point of inflection. The curve for 

 " p " has a continuously falling slope above 2 kilometers, that for 

 " r " has a rising slope (both in absolute value). Either rising slope 

 curve should show, by itself, a certain converging lens effect. 



5. Carrying out the differentiations indicated in paragraph 3 gives 



211 + „ (1515? -0.293 

 = 211?-f+fi515?_ 0.293 V^^f + P^^-'""""''^ 



d 



dh ' \ T I \ dh ' '^dhj T^ dh 



Three terms result, distinguishable, respectively, as due to the air 

 density gradient, the water vapor density gradient, and the tem- 

 perature gradient. 



As a typical and simple numerical example, we may select the 

 values of " p," " T," and " a " for h = 0, that is at the earth's 

 surface. We have then 



^ = - 1.85 X 10-^ «o = 1.372, 



dh 



po = 1.224 X 10-^ Ir = - 6.19, 



da 



dho 



0.405, To = 288, 



M = 28.6, 



and hence 



917 S 



R = -7^ 4t^ — 71^ = 14350 miles, 



390 + 262 - 12.6 



where the three terms in the denominator are: the air, water vapor, 

 and temperature gradient terms, respectively. It is evident that the 

 existence and distribution of the small amount of water vapor present 

 (1.37 per cent), adds very greatly to the effectiveness of the air itself 

 as a refractive medium. 



