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lefTened by three times the correfponding refra&ion 
taken out of any common table. 
Demonftration of the preceding rule. 
Let ZXY Tab. XVII. Fig. i. reprefent a fphe- 
rical triangle formed by great circles joining the ze- 
nith Z and the ftars X and Y. Refradtion adting 
in the vertical circles ZX and ZY wilL carry the 
Rar X nearer the zenith by a quantity Xb=z^y" x 
tangent of Z X, and the ftars Y towards Z by the 
quantity Y d= 57" x tangent of Z Y ; fo that the 
apparent diftance of the two ftars will be b d inftead 
of X Y or lefs than X Y, the true diftance, by the Sun of 
the two little fpaces Xa> Y e y terminated by the perpen- 
ndiculars b a and d e. The little fpace X a — X b X 
cofine of the angle ZXY (calling radius unity) = 
57 " X tang, of Z X x cofine of angle ZXY, or, 
by fpherics, = 57" x tang, of X P. (Z P being an 
arch drawn from Z perpendicular to the arch XY). 
In like manner the little fpace Ye — 57" X tang, of 
Y P ; and therefore X a 4- Y e or the total effedt of 
refradtion = 57" x tang. XP ~\- tang Y P. Let 
M be the middle of the arch X Y, and put the tan- 
gent of X M or Y M or X Y = t> and the tan- 
gent of M P, or the diftance of the perpendicular 
Z P from M the middle of the arch XY = By 
trigonometry, tang, of X P or XM-j-MPz:—, 
and tang, of Y P or Y M — M P — * — — ; the funi 
1 + t n 
of which, or tang. XP -f tang. YP 
t + n 
+ 
t — n 
2i + 2t n 
I — fn* 
2 t n 
l-fn x 
I + ri 
2 n 
1 -tn 
2. Now 
1 -Ytn 
2 t n 
