of the Atmospheric Absorption. 297 



so that, all the quantities being positive, by a known theorem 



L>L] ; and for the same values of A 1? A 2 , A 3 , this 



inequality is greater the greater the difference in the values 

 of the coefficients a^ a 2 , a 3 , 



But this is stating in other words that the true values found 

 by observing separate coefficients of transmission are always 

 greater than those found when we do not distinguish between 

 the radiations of which the light (or heat) of the star or sun 

 is composed, and also that the amount by which the true 

 values are greater increases with the difference between the 

 coefficients. 



We have stated above that the usual hypothesis makes the 

 coefficient of transmission a constant. It will be seen from 

 the above table, however, that it varies from one stratum to 

 the next ; that it is least when obtained by observations near 

 the zenith ; and that it increases progressively as we approach 

 the horizon. For, since a and b are each less than unity, each 

 of the sums Aa + Bb &c. in the above table is less than the 

 preceding. It is also evident that their rate of diminution 

 decreases as we approach the horizon, since 



Aa 2 -Aa 3 >Aa'-Aa\ Bb 2 -Bb 3 >Bb 3 -Bb\ 

 Hence 



(Aa 2 + Bb 2 ) - (Aa 3 + BV) > (Aa 3 + Bb") - (Aa 4 + Bb 4 ) . 



Consequently the difference between the numerators of two 

 successive ratios, such as 



Aa 3 + BZ> 3 Aa 4 + B6 4 

 Aa 2 + B£ 2< Aa 3 + B6 3 



is less than that of their denominators. In other words, 

 although both numerator and denominator decrease in suc- 

 cessive ratios, the ratios themselves increase progressively; 

 and a similar demonstration applies to the form 



Aa 2 + Bb 2 /Aa n + Bb n V=i 



/ Aa n + B^ Y 

 \ Aa + Bb ) 



Aa + Bb 



But these ratios are the coefficients of transmission in 

 question. 



Further, a simple inspection of the form of the expression 



Aa 2 - Aa 3 > Aa 3 - Aa 4 Bb 2 - Bb" > Bb z - B6 4 



shows that what is there demonstrated for two numbers and 

 two coefficients A, a, and B, b, is true for any number, even 

 infinite, which is the case we deal with in actual observation. 

 Phil. Mag. S. 5. Vol. 18. No. 113. Oct. 1084. X 



