501' Mr. Ivory on the Theory of the Astronomical Refractions. 



Investigation off x Qj • 

 We have 



Now the following formula is easily proved by differen- 

 tiating, 



/dx _^ __ 1 pdxc-'' m {\—e^f f*dxc~^ 

 ^^0 ^~j —^ — . —^— .J ——- 



m 1— e^— c-*A 



4 e^ 



all the integrals vanishing when x — 0. By extending the 

 integrals to ^ = »2 = 10, in which case A = 1 + e% the re- 

 sult will be 



/''» dx -^ _ i_ /» w d X c-^ 5 (1— g^)^ /^ mdxc-" 

 A "^ ^ " 2J A Y ' ? 'yo |_A 



_5_ 1— g^ + g'^^Cl+g^) . 

 ^ * g2 • 



and, by substituting this value, we shall have 



^~' 1/0 A 2^0 A 2^* g2 



/^nt edxc-^ , 5 1— g-*" 5 ,, ,„. 



The value of Q2 will [now be obtained in a series of the 

 powers of e by putting for the integrals the equivalent series 

 that have already been investigated. When this is done, the 

 three first terms will be as follows : 



Ki-g-'^-AO-i- 



+ (2 a,- I A, + 5 A,- -| A3- |(1 +g-)) . g 



+ {2a3 1 A3- |-A, + 5A3- f A,).g3. 



Upon substituting the exact values of Aj, A3, &c., the first 

 of these terms is zero : the other two are as follows ; 

 — 8 g-^ X g 



+ — g-*" X e^; 

 o 



the amount of which is very small even at the horizon; and, 



2 

 when multiplied by f = — , it becomes hisensible. These 



terms being neglected, we may assume 



Q2 = C5 g-^ + C^e' + CggS + &c. ; 



