542 A CATALOGUE OF STANDARD STARS. 
taking the quadrant of £, so that m shall be positive. ; 
We shall readily find (Fundamenta, p. 300 (9) ) 
sin à — cos 8 sin Ai — sin 6 cos Ai cos (a! + 1 — z^), 
sin $? + m? = sin 8? + m? sin Y + m? cos  — 1. 
Hence, 
m? — cos ð’; m (always positive) = cos 4, 
. , __ sin 6 sin (a! + 4! — 2") 
=i cos à : 
and 
4 o! cos à! = Ja cos d cos ¿+ 40 sin E, 
45 = — do cos à sin 5 - 40 cos & 
Hence, again, 
A a! cos Y = (n +2) cos ð cos £+ (++ =) sin E 
30 30 : 
49 =-— nt 22) eos à sing + (0+ E cos §. 
e 30 30 
If aa, d be of the nature of corrections to the assumed place, and 4a’ cos 0, 
4 à' be of an opposite character, — that is, be * computed — observed," — we must 
necessarily change the sign of one term; or, what is the same, transfer Je cos 9, 
A ð to the right members of their proper equations. 
We have now the form of the equations of condition; perfectly rigorous. It is, 
however, obvious that in most cases perfect rigor is not necessary. For example, for 
stars near the equator no one would think of considering the declinations of 1755 as 
so erroneous that a perfectly accurate reduction of AR. to 1855 could not be effected 
by its means. 
The rigorous course has here been followed for the four stars for which the value of 
sin £ was the largest. These stars are 4 Ursee Minoris (Bode), 51 Cephei Hevelii, 4 
Urs Minoris (Bode) and 20 Urse Minoris (Bode). 
For the remaining stars of our polar list, a value of «, « was deduced from the 
observed declinations, assuming that y, y” were — 0. Substituting these values in 
the equations derived from the right-ascensions observed, the values of y, y” were 
now obtained; employing in both cases the method of least squares, as will be 
further explained in its place. 
These values of y, y were now substituted in the declination equations, and a new 
set of values of «, “, differing by very small amounts from the former, were obtained. 
