222 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



wherefore 



l0-\-d. 10 = 10(1+—) 



• /« «(!+«) ^^Y'^ 1 1 -e^ dS \ 

 -^^^^' x/Ti ' i \2 ~'Y' l+e^' de ^) 



\ , ' A «(1 +«) /^« di\ ^ „ 

 + sm ^ • .-— . • ( r I . X Q,. 



If p denote the observed height of the barometer, reduced to the fixed temperature 

 of 50° of Fahr. ; and r the temperature of the air on the same scale ; then, /3 = ^j 



30 — ;? 



30 



These values being found, if we put 



^57 ^480><\ 2(1+0 4-2X^1^, 



T = sin ^ X \/— . 



V 52 



fe = sm^X-^;7^X-^; 

 the expression of the mean refraction with its correction will be as follows, 



^0 -{- d.h0= ^ _ ^l^ _ ^^y §^ -T .(t ^ 50) - b{30 •- p). 



The first term of this expression is the mean refraction corrected in the manner 

 usually practised by Astronomers. If we assume that the temperature of the mer- 

 cury in the barometer is the same with that of the air, this term will be equal to 

 1 1 p 1 p 



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

 "^ 10000 



c = -002183, 

 the new factor being added to compensate the expansion of the mercury. Two sub- 

 sidiary tables are given for computing this part : Table II. contains the logarithms 



^^ TT — ( — 50) ^^^ ^^° ^^ either side of the mean temperature 50°, negative indices 



being avoided by substituting the arithmetical complements ; and Table III. contains 

 the logarithms, or the arithmetical complements, for all values of ^ from 31 to 28. 

 The coefiicients, T and b, of the other two terms vary with the distance from the 



