216 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



wards be discussed, and its value investigated, in order that it may be taken into ac- 

 count, if this should be found necessary, in the progressive improvement of the theory. 

 When the term/' Q3 is left out, the expression of the refraction will be 



hd = sin^ X ^^^^~f ' (Qo + XQi -/Q^) : 



and if the equivalent series be substituted for the first two terms, and the series 



2 

 for Q2 be multiplied by/ = -^, the result will be 



5^ = sm^X-^7f^x|^ 



+ 0-821921 .e3 

 + 0-552697 . e^ 

 + 0-302296 . e7 

 + 0-132123 .e9 

 + 0-043365 -e" 

 + 0-007883 . 6^3 

 . — 0-002040 . ei5 



— 0-002930 . ei7 



— 0-00 1842. ei9 



— 0-000893 . e2i 



— 0-000371 . e23 



— 0000 139.^25 



To bring this formula to a form more convenient for calculation, all the coefficients 

 must be reduced to seconds. The negative terms are all very small, never amounting 

 to so much as 6", and of no account whatever, except the apparent altitude be equal 

 to 2° or less ; it will therefore be proper to separate these terms from the rest, repre- 

 senting their sum by the symbol V (0). These things being attended to, we have, in 

 the first place, this formula for computing e, viz. 



log . tan ^ = log . sec . ^ + 19-2067840 — 20 : e = tan -|- 



Next, reducing the arcs to seconds. 



