WHICH REFLECT LIGHT. 319 



On account of the assumed smallness of w — 1, we can simplify 

 this. This causes the angle of reflexion i—i' to be always small, 

 and we can therefore set cos^(i— i') = l, and neglect sm^{i—i') in 



comparison with 1 ; but not so with the fraction . J). — rr-. for 



sm^(2 + 



this, how small soever i—i' may be, at least for the limiting 

 value i=90°, has the value of 1. Applying this, the foregoing 

 formula becomes 



^ . . . ..r^sin^li—i') sin'* (i — i') . g,. .„"| . , , 



2sm^cos^c?^ 2- . ^). . ., ; —-^^4^ — J —sm^U—t') . (11a) 



L sm^(z + *) sm*(^ + ^') ^ ^J ^ ^ 



As this expression merely denotes the portion of the reflected 

 light which corresponds to one elementary zone, it must be in- 

 tegrated between the limits i=0 and i=— , which may be 



effected when the angle i by means of the equation sini'= 



is eliminated. The quantity of light reflected from the entire 

 sphere is thus found to be 



127i2 J 11^) 



Applying this formula, which for the sake of brevity we shall 

 call M, to the total quantity of light A which shines through 

 the distance o-, and remembering that the loss suffered in this 

 distance must, according to the notation already made use of, 

 be expressed by AAo-, we obtain 



^=- (12«) 



In a similar manner the coefficient of dispersion k must be 

 determined. For this purpose let us return to the considerations 

 by which the meaning of this coefficient w^as fixed, and deter- 

 mine how much luminous intensity passes through the element 

 EF= c?y (Plate V. fig. 1) while the rays are passing over the distance 

 cr through the atmosphere, the original distribution of the lumi- 

 nous intensity in the firmament being expressed by the equation 



This investigation also will be simplified by the assumption 



