NO. 8 SAMUEL PIERPONT LANGLEY — ABBOT I3 



" Then will 



Aa + Bb Aa- + B/r Aa' + Bb' Aa^ + Bb* 



A + B "" Aa+Bb ^ Aar + Bb' ^ Aa' + Bb' ^^ ^' 



and 



Aa' + Bb- 



< 



( Aa' + Bb' Y ( Aa' + Bby 

 \ Aa+Bb / ^V Aa+B& / ^ 



Aa+Bb 



" 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 Aa+Bb is the sum of the two kinds of radiation 

 as observed after absorption by one unit stratum (sec.^=i) by the 

 photometer, or actinometer; Aar + Bb- is the sum of the radiations 

 observed after absorption by two strata (sec.^ = 2) etc. ; but we are 

 here supposed to independently know the really dual constitution of 

 the radiation, which the photometer or actinometer does not discern. 

 According to the usual hypothesis, the coefficient of transmission, 

 which is the quotient obtained by dividing the value after n absorptions 

 by that after n — i absorptions, or more generally that from the 

 expression 



Value after n absorptions 



r 



\ Value after m absorptions 



is a constant. It is in fact not a constant, as we shall prove later ; but 

 we shall first show that, if we proceed upon the ordinary assumption, 

 the value obtained for the original light of the star before absorption 

 will in this case be too small. For, if we observe by a method which 

 discriminates between the two radiations, we shall have, if we sepa- 

 rately deduce the original lights from our observation of what remains 

 after one and again after two absorptions, the true sum 



Aa- Bb- 



while if we observe by the ordinary method, which makes no discrimi- 

 nation, we shall have the erroneous equation 



j(Aa+B&r 

 ^ + ^- Aa^ + Bb^ 



which is algebraically less than the first, or correct value, for the 



expression 



(Aa)2 (Bby (Aa + Bby 



' T-> 7 O -^ 



Aa^ ' B&2 ^ Aa--|-B&2 

 readily reduces to the known form 



a^ + b^>2ab. 



