214 MR. IVORY ON THE THEORY OP ASTRONOMICAL REFRACTIONS. 



all the integrals vanishing when x = 0. By extending the integrals to ^ = m = 10, 

 in which case A = I + e^, the result will be 



Jo A^^ — '2Jo A Q. ' e^ Jo A + 2 ■ e^ 



and, by substituting this value, we shall have 



Q2 — 2 ej^ ^- ^^J^ ^ ~ q ' e^ 'Jo A 



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 : 



+ (2ai-|-Ai4-5Ai-4A3-4(l+c-'"))-e 



+ (2 ag - 4 A3 - 4 Ai + 5 A3 - -|- As^ • e3. 



Upon substituting the exact values of Aj, A3, &c., the first of these terms is zero : 

 the other two are as follows : 



- 8 c- "* X e 



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



2_ 

 9 



2 

 f z= -Q-, it becomes insensible. These terms being neglected, we may assume 



Q2 z= C5 e^ + C; e7 + Cg e9 + &c. ; 

 and we shall find 



C5 = 2 ag — ^ A^ — -^ A A3 



C; = 2 a; — -^ A; q A A5 



The numerical coefficients will now be obtained : 



A2Ai = — •0802325 

 A2 A3 = + -0403433 Cg = "059337 



