194 Mr. Nixon on the Measurement by Trigonometry of the. 

 the (positive) refraction r 9 will be d — r and e + r. Hence 

 (e -fa) -rf ^ or a '- J: — ^ w jji De tne re f rac tion * ; but the an- 

 gular difference of level maybe obtained without previously 

 ascertaining the refraction or having given the contained arc, 



DEE' being equal to —^ — , or to -^- — . 



When the instrument, from some imperfection in its con- 

 struction, gives the elevations in excess or defect by an un- 

 known but constant quantity y, then will e* be equal, in the 

 former case, to (e + y -f- r), and d! to (d — y — r); but in the 

 latter case we shall have e 1 = (e — y + r) and d'= (d + y — r) ; 



whence — — , or — ^— , will be the correct angular difference 



of levelf, and 2 , or < L^i_Z — * the refraction plus, or 



minus the constant error of the instrument, according as the 

 elevations are in excess or defect J. Were the refraction a 

 constant ratio of the arc, its value, unaffected by the instru- 

 mental error, might be determined from reciprocal observation 

 on arcs differing in extent. Let (r ± y) be the observed re- 

 fraction for the arc a, and (V + y) that for the (much) greater 

 arc a! ; then (r' ± y) — (r ± y) will be the true refraction for 

 the arc («' — «). 



Unfortunately for the accuracy of trigonometrical measure- 

 ments, it will frequently occur, especially on low grounds with 

 a cloudless sky in the spring, that a thin stratum of air in con- 



* When the observations are not reduced to the ground, the refraction, 

 calling h the angle subtended by the sum of the heights of the eye at the 



two stations, will be — , or 2 ■— . 



d' 4- e 

 f When the height of the eye is the same at both stations — - — , or 



<f— d 7 



— , will still give at once the angular difference of level /; for 



a h. , C 



whence d -f- «' = 2 / ; 

 and 



or -£- + r — v / = D' ; whence d' - D'= 2/. 



2^2 



J When y is known, and the observations are not reduced to the ground, 

 the refraction will be equal to 



(a + h + e' ± 2y) - <£ (a + h + 1y) - (<f + DO 

 2 , or to — 2 . 



tact 



