DR L. BECKER ON THE SOLAR SPECTRUM. 121 



Since Z has been supposed to be large and /3 = 745747 in Bessel's hypothesis, 

 the correction is very small indeed. This holds also for smaller values of Z, because 



; . ,- ErF , makes the first term # in — cotg 2 Z , vanish. 



(n+rn)J2p «/u+m 



For small values of Z we can develope dF (4) according to powers of tg 2 Z. Neglecting 



all terms multiplied by s 3 and a 2 we obtain 



cosZ/Sn + l 1 ^(w+l)^"' * Un + 1)/T (w+l) 2 /^ w+2/ 

 , 47 / 3_ _3a 2^ + 3 \ ) 



For the zenith the formula becomes 



P «JSff ( 1+ «i + l (11 ). 



/3 71+1 (_ + l)/3 j v ' 



With Bessel's values of -5 and ^ we find for the oxygen of our atmosphere 



F = 5572 feet if n be equal 0, 

 F = 579 feet if n be equal 1 ; 



or, in words, the absorption produced by the oxygen of our atmosphere in the zenith is 

 the same as that of a column 5572 feet in length of oxygen under a pressure of 

 1 atmosphere, if the absorption be proportional to the density ; and of 579 feet, if it 

 be proportional to the square of the density. M. Janssen finds 1660 and 172 metres 

 respectively, by employing a coefficient which Ramont had computed from the heights 

 of mountains determined by barometric and trigonometric measurements.t 



The formulae (7), (8), (9), (10) enable us to compute the absorption in any zenith- 

 distance, but for our purpose we may dispense with the corrections given in (8) and (9). 



For n = they admit of great simplification. This special value makes the expression 

 within the bracket of (7) identical with that in Laplace's and Bessel's formula of 

 refraction. Hence we obtain 



1 a l-a, 7 



1 sinZj8 ru a 



in which SZ denotes the astronomical refraction, or in units of F in the zenith (see (11)) 



A- 1 ™ < 12 >- 



This result might have been immediately deduced from the fundamental equations. 



* Laplace, ibid., No. 5. f Laplace, Traite de mdcanique c&este, Tome iv. livre x. chapitre iv. 



