north polar distances of the principal fixed stars. 59 
to k, even differing 1", by making corresponding changes in 
g, so that the problem partakes too much of the nature of an 
indeterminate one. Thus the advantage apparently gained 
by large refractions, is lost by attendant inconveniencies. 
In my investigation, there is only one unknown quantity, 
but then I have much smaller quantities to work with. 
Theory gives as far as about 7 6°, whatever be the law of 
variation of density in the atmosphere. 
* The mean refraction (r) = k tan. % 77S7 0 )» 
% being the zenith distance. 
By a table of refractions, or by the pole star, and a star or 
stars more remote, k is easily obtained nearly. Then if the 
true value be k + d k 
dr= dk tan. z ( 2 ) sufficiently exact. 
Let A and B be the observed zenith distances of a circum- 
polar star, (considering B negative when south of the zenith ) 
above and below the pole, R & R' the refractions exactly com- 
puted by the formula ( 1 ) , k being the approximate constant 
of refraction. 
Then by ( 2 ) 
A-fB + R + R' + dk tan. A -f- dh tan. B 
' 2 
Hence, if C represent the mean co-latitude thus determined 
by circumpolar stars remote from the pole, and N that by 
stars near the pole, we obtain an equation of the form 
C-\-mdk — N-\-ndk 
m — n 
In this investigation the Z. D. of the stars remote from the 
Pole, should not be greater, when below the Pole, than about 
* Transactions of the Royal Irish Academy, Vol. iz. 
