S. P. Langley—Atmospheric Absorption. ae 
decreases as we approach the horizon, since 
Aad’— Aa’ > Ad’—Aa* Bo’?— Bs’? > BL*— Bd‘. 
Hence 
(Aa’= Bd’) — (Aa*— Bod’) > (Aa’ + Bd*)— (Aat + Bd‘). 
Consequently the difference between the numerators of two 
‘successive ratios, such as 
Aa’ + BB’ Aa‘ + Bd 
Ad +Be S Aas Be 
is less than that of their denominators. In other words, 
although both numerator and denominator decrease in success- 
ive ratios, the ratios themselves increase progressively, and a 
similar demonstration applies to the form 
Ad +Bo? _ /Aa’+Bb"\ 
Aa+Bs ~ \Act+Bo)” 
But these ratios are the coefficients of transmission in question. 
gain, a simple inspection of the form of the expression, 
Aad’— Aa’ > Aa’— Aa‘ Be’? — Bo? > BS'— Bo 
shows that what is there demonstrated for two numbers and 
two coefficients A, a, and B, 6, is true for any number, even — 
Infinite, which is the case we deal with in actual observation. 
__It is then universally true that when the numbers are pos- 
SUVO, ANG GD Oe eo. proper fractions A 
Aa*t1 + Bott! 4+Co"'t1 4+ Dd*ti+ 2... mt 
Aa +Bé" +Cc® +Dd" + 
—_Aatt® 4 Bort? 4. Cert? 4 Dd"t24 2... 
Sagi + BS tt = Cett + Dd*tt + 2 ee 
and hence universally true, that when the separate coefiicients 
of transmission are positive and less than unity (as is the case 
in Nature), the general coefficient of transmission in the cus- 
tomary exponential formula is, 
1) never a‘ constant, and (as determined from the customary 
formula) 
(2) always too large, 
(3) always larger and larger as we approach the horizon. 
) The original light or heat of the heavenly body as found 
by the photometric and actinometric processes, and the form- 
ulze, in universal use, 7s always too small, a conclusion which | 
we have just reached by another method. 
The above demonstration does not tell us in how greata 
degree this coefficient is too large, and for aught we have here 
yet shown, the error may be practically negligible. : 
Since the method ordinarily employed demonstrably gives 
too small results, the burden of proof might seem to rest on 
