of t lie Atmospheric Absorption. 



295 



as it depends on this formula, always, and invariably, too small. 

 The demonstration may be put in an extremely simple form, 

 but I am not aware that it has been elsewhere given, though 

 it was indicated in the proceedings just cited. 



Let us first suppose the radiation of the heavenly body to 

 be really composed before absorption of two portions, A and 

 B. Let A have a special coefficient of transmission (a), 

 and B another, special to itself (b). Then, if we assume 

 (still for considerations of convenience only) that each of 

 these portions is, separately considered, homogeneous, we 

 may write down the results in the form of two geometrical 

 progressions, thus : — 



Table I. 



Original 

 radiation. 



Ratio. 



Eadiation 

 received after 

 absorption by 

 one stratum. 



By two 



strata. 



By three 



strata. 



By four 

 strata, &c. 



A 

 B 



a 

 b 



Aa 

 Bb 



Aa 2 

 Bb 2 



Aa 3 



B63 



Aa 4 

 B5± 



A+B 





Aa+Bb 

 =(M) 



Aa 2 4-B& 2 



= (N) 



Aa 3 +Bb 3 

 = (0) 



Aa±+B6 4 



Then will 

 Aa+Bb A^ + Bff^ 



Aa 3 + B^ 



A + B Aa + B6 Aa 2 + B6 2 



^ Aa' + BV ^ s 

 < Aa> + Bb> <&Xi ' 



and 



Aa 2 + B& 2 JAcl 

 Aa+Bb 



/Aa 3 + B£ 3 \WAa 4 + BrU 

 {-A^fBb ) <{ -AaTBb) <&C - * 



The fractions here are the coefficients of transmission, as 

 deduced from observations at different zenith-distances. They 

 evidently differ, and (as will be shown) each is larger than 

 the preceding. 



In the above table Aa + B5 is the sum of the two kinds of 

 radiation as observed after absorption by one unit stratum 

 (sec f=l) by the photometer or actinometer ; Aa 2 -f Bb 2 is 

 the sum of the radiations observed after absorption by two 

 strata (sec ?=2) &c. ; but we are here supposed to inde- 

 pendently know the really dual constitution of the radiation, 



