Mr. Ivory on the Theory of the Astronomical Refractions, 505 

 These values being found, if we assume 



the term A x C"'" X e, which is insensible, being omitted, we 

 shall hnve 



Bg = A (ag — Ag) = 0*021866 



Bg = A (ag - Ag ) = 0-04596 1 



B7 = A (87 - A7 ) = '055 760 



B9 = A (ag — Ag ) = 0-049829 



Bii = A (a,! — All) = 0-036064 



Big = A (ai3 — Aig) = 0-022242 



B15 = A (ai5 - A15) = 0-012064 



Bi7 = A (a]7 — A17) = 0-005878 



Big = A (aig ~ A19) = 0-002610 



i^2i = ^ (%i - K\) = 0-001067 

 B23 = >^ (agg — A23) = 0-000405 

 B25 = ^ (a25 - A25) = 0-000144 

 By making cos 9 = 0, ^ = 1, we shall have, for the approxi- 

 mate value of the part of the horizontal refraction depending 

 on A Qi , 



^^^i^X 0-253891 = 184"-50. 

 V 5 I 



If the integrals be taken from ^ = to ^ = 00, the same 



quantity will be 



= 184"-56. 



Between the two limits, the exact quantity obtained by inte- 

 grating from a; = to x =z m = 10, must lie; so that the 

 error of the series is of no account. 



It may be proper to make an observation here, which ap- 

 plies generally to the kind of integrals peculiar to this inves- 

 tigation. The first term of A Qj , viz. Bi e = A 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 refrac- 

 tions will vary with the height, and will not agree with the ob- 

 served quantities ; if a considerable number be taken, as eight 

 or ten, or any greater number, that is, if the atmosphere ex- 

 tend 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 elevation of fifty miles, more or less. 



