60 BULLETIN OF THE UNIVERSITY OF WISCONSIN 



I. -A SIMPLE BUT ACCURATE EXPRESSION FOR THE 

 ATMOSPHERIC REFRACTION. 



Bessel's expression for the refraction ' 



A A 



R = a ft y tan z 



is commonly employed for all accurate computations of the 

 refraction, and when so employed requires that the five 

 quantities, oc, ft, r. A, A, shall be interpolated from specially 

 prepared refraction tables. It is the purpose of the present 

 paper to so transform this expression that the refraction 

 may be computed without recourse to these tables. 



Since the refraction admits of development in terms of 

 the odd powers of tan z, we may write for the mean re- 

 fraction : 



R,^ = a tan z = a^ tan z — a^ tan^ z etc. 



o"! f 1 ^ tan'*z\ tan z (approximately 



The Pulkowa Refraction Tables are presumably the most 

 accurate ones available at the present time, and from these 

 tables I find: 



a^ = 57.584 a, = 0.0640 



If with these values we compute 





tan^z 



and compare it with the tabular values of « we shall find 

 the following satisfactory agreement: 



z 0° 20° 40° 60° 75" 



W B » M M 



Tabular a 57.586 57.577 57.538 57.386 56.694 



Formula 57.584 57.576 57.537 57.391 56.693 



The quantity a is a complicated function of the zenith 



•' Tab. Reg., LXII. 



