MR. IVORY ON THE THEORY OP ASTRONOMICAL REFRACTIONS. 213 



B5 = X (as — A5) = 0-045961 



Bj = X (a; — A7) = 0-055760 



Bg = X (ag - Ag) = 0-049829 



Bn = X (an — A^) = 0-036064 



Bi3 = X (ai3 — A13) = 0-022242 



Bi5 = >^(ai5"-Ai5) = 0-012064 



Bi7 = X (ai7 - A17) = 0-005878 



Big = X (ai9 - Ajg) = 0-002610 



B21 = X (a2i — A21) = 0-001067 



B23 = X (a23 — A23) = 0-000405 



B25 = X (a^5 - A25) = 0-000144. 

 By making cos d = 0, e = 1, we shall have, for the approximate value of the part of 

 the horizontal refraction depending on X Q^, 



a(l + a) 



\—. X 0-253891 = 184"-50. 



Vol 



If the integrals be taken from a? = to j? = cvj, the same quantity will be 



X X « (1 + a) /''"'^°;^-^'^' = "(■ +/1^~- . X (^il - I) = 184"-56. 



Between the two limits, the exact quantity obtained by integrating from .r = to 

 X = m = IO3 must lie ; so that the error of the series is of no account. 



It may be proper to make an observation here, which applies generally to the kind 

 of integrals peculiar to this investigation. The first term of X Q^, viz. Bj e = X c~ *" . e, 

 which is rejected, varies with the height of the atmosphere. If a small number be 

 taken for m, that is, in low atmospheres, the refractions will vary with the height, and 

 will not agree with the observed quantities ; if a considerable number be taken, as 

 eight or ten, or any greater number, that is, if the atmosphere extend forty or fifty 

 miles or more above the earth's surface, the refractions will not be sensibly different 

 from what they would be in an atmosphere of unlimited height. The invariability of 

 the refractions concurs with other phenomena to prove that the air reaches an eleva- 

 tion of fifty miles, more or less. 



Investigation off X Qz- 

 We have 



e 



Now the following formula is easily proved by differentiating, 



/ dx _ ^ 1_ p dxc-'^ m_ (1 - e^Y fdxc-' m \ — e^ — C ^ ^ 



