Light in the Atmosphere, 



43 



homogeneous, the actual quantity of light which arrives, which 

 may be designated by S, can be determined by the following 

 equation, a 



S=sin3.e i&S, (1) 



in which e is the base of the natural logarithms, a the above- 

 mentioned coefficient which depends on the degree of transpa- 

 rency of the atmosphere. 



The difference, a 



sin 3— sin d . e~ sin^ 



represents the loss which the direct solar light has experienced 

 by the action of the atmosphere. Assuming now that this quan- 

 tity of light is withdrawn from the direct solar light by the cir- 

 cumstance that it is reflected from any constituents of the atmo- 

 sphere, it must (either directly or after experiencing several 

 reflections in the atmosphere) partially reach the earth as diffused 

 light, and partly be radiated into space. The question is, what 

 fraction of this quantity of light reaches the earth ? and in an- 

 swering this question, the nature of the constituents of the 

 atmosphere which effect the reflection must be taken into account. 

 We will denote by Z this fraction, which varies with the position 

 of the sun, and is therefore to be regarded as a function of $ ; 

 the quantity of light which falls upon the unit of surface as dif- 

 fused light from the sky, and which may be designated by H, 

 is determined by the equation 



H=sin$(l-e"iiib")z. . .... (2) 



Dividing the equations (1) and (2), we obtain the desired ratio 

 between the direct sunlight and the diffused light of the sky ; 

 that is, a 



- — r4- ...... (3) 



H 



l-e~ 



sinS- 



The magnitude Z depends mainly, as has already been stated, 

 on the nature of the reflecting constituents. Such differences 

 of the coefficient a as can occur in bright weather exert so small 

 an influence on the magnitude Z, that in an approximate calcu- 

 lation this may be regarded as independent of a. Assuming 

 that the reflection is occasioned by water- vesicles, the following- 

 values for Z are obtained from my previously published calcu- 

 lations : — 



3 



20°. 



25°. 



30°. 



35°. 



40°. 



50°. 



60°. 



z 



0-575 



0606 



0632 



0-654 



0-673 



0-701 



0-721 



