[ 26 7 ] 
is the tangent of double the angle whofe tan- 
gent is t n, and t n or the product of the tangents 
of 4 - X Y and M P, by fpherics, is equal to the 
product of the tangents of half the fum and half 
the difference of the zenith diffances Z X and Z Y ; 
whence n - * s ec l ua i t0 ^ tangent of double arch 
the firft found by the rule. Alfo arch the fe- 
cond found by the rule being by fpherics — M P, 
whofe tangent is reprefented by 72, and being 
2 > Tl 
by trigonometry equal to the cofecant of double the 
arch whofe tangent is therefore — ^ = cofecant 
of 2 MP or double arch the fecond. Whence 
the rule is manifeft; namely, that Ha q-' Y e, the 
total effeCt of refraction in contracting the apparent 
diffance of the two ftars = 57" x tan. X Pq-tan. YP 
= double 57" or 114" x tang, of double the frit 
arch x cofecant of double the fecond arch. Q^E.D. 
REMARK. 
When the perpendicular arch ZP, let fall from 
the Zenith on the arch X Y, falls without the trian- 
gle Z X Y, the effeCt of refraCtion in diminifhing 
the apparent diftance of the ftars X, Y is the differ- 
ence of X a and Y e : but the rule being general, 
gives always the fum or difference, which- ever it be, 
which is a great advantage* and removes all grounds 
of ambiguity in the correction of refraCtion ; as the 
the total effeCt thereof always diminifhes the dift- 
ance of two ftars from each other, however they are 
pofited. 
Mm2 
A 
