of the Atmospheric Absorption. 299 



in substantial agreement (within three or four per cent.) from 

 independent observations made at very different altitudes of 

 the sun or star, the proof of this last statement may well be 

 demanded, and I proceed to give it. 



(It is admitted that the original energy is divisible into an 

 unlimited number of rays, each characterized by its own 

 coefficient of transmission, and that these coefficients have 

 values varying from zero to unity.) Let this original energy 

 of the heavenly body be X, and let us denote, as in Table I., 

 the observed energy after one absorption by M, after two by 

 N, after three by 0, &c. 



The assumption of those who admit the existence of an 



error but maintain that it is negligible, is that, since we do 



know that : N = N : M very nearly, therefore N" : M = M : X 



very nearly; or, in other words, that if we can prove that 



N 2 . 

 M=-y- within three or four per cent., we are entitled to 



assume that there is only a like small error in writing 

 ^ M 2 



We can make the fallacy of the preceding assumption most 

 obvious by first considering a particular case. Let the ori- 

 ginal energy be divided into any number of parts A, B, C, &c, 

 each with its own coefficient a, b, c, &c, so that 



Aa + Bb +Cc + Vd + Kk +U &c. = M, 



Aa 2 + Bb 2 + Cc 2 + T>d 2 + KP + LZ 2 &c. = N, 



Aa 3 + B6 3 + Cc 3 + Dd 3 + KP + LZ 3 &c. = 0, 



&c, &c, . &c. 



We have only to assume that k, I, &c. are sufficiently near 

 zero (so that K, L, and all other rays affected with such 

 coefficients sensibly vanish before they reach the observer) to 

 see that the only quantities sensible to observation are those 

 with relatively large coefficients as A, B, C, D, &c; so that 

 now 



Aa +B6 +Cc +T)d + &c. = M, 

 Aa 2 + Bb 2 + Cc 2 + Da 72 + &c. = N, 

 Aa 3 + B6 3 + Cc 3 + Dd 3 + &c. = 0. 



From these values M, N, 0, &c. we can never estimate the 

 amount of the extinguished rays K, L, &c, since these do not 

 enter into the observed values by any amount sensible at all. 

 Now to the rays A, B, C, D, &c, which remain, and to 

 their coefficients, we may evidently assign any values con- 



X 2 



