Mr. Ivory 07i the Theory of the Astronomical Refractiom. 95 



The first term of this expression is the mean refraction cor- 

 rected in the manner usually practised by astronomers. If 

 we assume that the temperature of the mercury in the baro- 

 meter is the same with that of the air, this term will be equal 

 to 



1 1 p _ 1 jp_ 



1+^(t— 50) '~ T— 50 '30" 1+c(t-50) * 30' 

 '*' lOOOO 

 c — -002183, 



the new factor being added to compensate the expansion 

 of the mercury. Two subsidiary tables are given for com- 

 puting this part: Table II. contains the logarithms of 



:; — r for 30° on either side of the mean temperature 



1 +c (t — 50) ^ 



50^, negative indices being avoided by substituting the arith- 

 metical complements ; and Table III. contains the logarithms, 

 or the arithmetical complements, for all values of y from 31 

 to 28. 



The coefficients, T and h, of the other two terms vary with 

 the distance from the zenith ; and they can be computed in 

 no other way than by reducing them to series of the powers 

 of e. By substituting for X Qj, the equivalent series already' 

 known, we immediately obtain 



h = sinS.^^^^iL^ 1.163^3+65^^ + 67^7+ &c. "I . 

 ^52 30 \^^^ 7 j 



Further, by expanding S and its differential, the expression 

 of T will take this form, 



T = sin9 . "^iit?! .-^ .^Gg ^ + G7^' + G9^ + &c. "I ; 

 'Z 5 z 480 L J 



and we shall have 



G3 = Ai~ A3 + 2 63 = 0-24.36 



G.5 = — Ai + 3 A3-2 A5 + 2 Bg = 0-4523 



G- = Aj— 3A3 + 5A5— 3A7 + 2B7=0-4705 



Gg = -Ai + 3A3-5A5+7A7-4A9 + 2B3 = 0-3502 



G,i = A1-3A3 + 5A.5— 7A7+9A9-5A11 + 2B11 =0-2092 



Gi3=-A, +3A3-5A5 + 7A7-9A9+iiA„-6Ai3 + 2Bi3 



= 0-1050. 



The series for T and h being now known, the coefficients of 



