ALTAZIMUTH SOLAR OBSERVATIONS. 339 
It has also already been remarked that the vertical contracted 
semidiameter may be used for the length O P, vide p. 337, this 
section, and it may be noticed that the maximum value of y, 
assuming x to be 20°, and the altitude to be 5°, is only about 
52’, so that the error of assuming its cosine also to be unity is 
only about 334-5. This would involve an error of 0-’33 in the 
value of Y, and nothing sensible in that of X,! consequently the 
equations (38) may, for the system of wires represented in Fig. 3 
always be put in the form 
X=8;, (sin x - cos x cot y) +9, COS X COSEC Y......ece eee \ ‘ 
Y=S, cos y sin é cosec y — S, (sin € cot y + cos é)...... 
the plus sign in the last written term being taken for the case 
illustrated in the figure, and the minus sign when P is above the 
line Im. These equations do not involve an error of 0°01. 
When the altitude is 15° or more, the substitution of unity for 
cos y will not involve an error of 0-’01 in the value of Y. Hence 
we see that for any altitude from 15° upwards, the image of the 
sun may be assumed to touch the wires at points determined by 
letting fall perpendiculars thereon from the image of the sun’s centre. 
And this may always be assumed for small theodolites, or for such 
as do not read to less than 1”. 
If the wire IP be perfectly ‘at right angles to the vertical 
through J, 
X=8,,and Y=S, cos y cosec y — 8, cot y (40) 
The desirableness of securing exact adjustment is very obvious on 
Comparing this last formula with (38) or (39). 
In Fig. 3 let Im’ represent an almucantar drawn through I: 
Seen. in applying formula (7) for its computation, we may take 
mm’ = m say, and 
m = $(S cosec y — 8, cot y)? cot z (w) 
For y= 70° this correction will amount only to 1°10 cot z, a 
formula by means of which mm’ may generally be computed for 
‘If the I P wire be horizontal the value of the S; term in X is zero. 
