( 587 ) 



and integration 



(Onr-l)lg(l—& n ) = lg{l—U>) .... (VII) 



in which 1 — <x> represents the ratio of the densities in the two 



horizontal planes. 



If we substitute n -f- 1 for n\ we can find in table II the tem- 

 perature for the two planes and hence also &„; as //„ and y n +\ are 

 also known, we can derive from (VI) the value of c„ and we can 

 deduce from (VII) the ratio of the densities in those planes. By put- 

 ting for n successively 0, 1, 2, etc. we can construct a table con- 

 taining the densities of the air, D lt D. 2 , ZJ S etc. at the height of 

 1, 2, 3, etc. kil. above the surface of the earth, the density at the 

 surface being unity. 



It is easy to derive trom this table the height of a layer of a 

 given density d. If d <^ D n and ^>/)„_|_i, the layer must be situated 

 between n and u -f- 1 kil., and we only want to know in which 

 manner, within this kil., the density varies with the height h above 

 the lower plane. 



We may assume : 



d 

 D n 



For h = 1 kil., d = Z) n + i, hence a = — ly " . 



a being known, we may determine for each value of <l , h and 

 also y. By substitution in (I) we tind then for each value of to the 

 value of ds. 



7. Now we are able to form the differences of d.s after the theory 

 of Ivory and after the table of temperatures II, for values of co which 

 increase with equal amounts, and then determine the whole diffe- 

 rence of the refraction for both cases. 



For great values of z and small values of y and to the coefficients 

 of du* in (I) will become rather large, which derogates from the 

 precision of the results. 



This will also be the case when the differences of the successive 

 values of to are large; small differences are therefore to be preferred, 

 but they render the computation longer. 



Both these difficulties can be partly avoided it', according to 

 Radau's remark, we introduce |/<o,as a variable quantity instead of to ; 

 the value of d$ thus becomes: 



