ALTAZIMUTH SOLAR OBSERVATIONS. 335 
CN can never be greater than about 16’ each, it is evident that 
the relation between C and I may be ascertained with sufficient 
rigorousness by treating the problem, so far as that relation is 
concerned, as plane instead of spherical. The dimensions of the 
elliptical image, together with the magnitudes of the angles at I 
and J, and their relation to a vertical drawn through this latter 
point, admit of a complete determination of the quadrilateral 
IPCQ. It may be remarked that for the system of wires illus- 
trated in the figure, the angle QI P is generally about 110°, and 
QI makes an angle of about 20° with the vertical. IP should 
be at right angles to a vertical passing through J: with ordinary 
care in adjustment an error of 1° in that respect will rarely be 
found. If P p and Qq be normals to the curve, the angle at p 
will consequently be between 89° and 91°, and at q about 20°. 
It is always intended that the intersection J shall be coincident 
with that at I: this is never perfectly realized so that the inter- 
sections must be treated as non-coincident. 
An expression will hereafter be required for the difference of 
direction between the normals and the radii vectores from P and Q. 
For convenience put a = S, the semidiameter from the ephemeris 
for the date of observation, corrected for augmentation and hori- 
zontal refraction : and put also b = S - c = S, viz. the vertical 
- semidiameter similarly corrected ; then designating the angles at 
C, P,Q, p and q as follows,— : 
PpN'=£,PCN'=£¢,CPp = £—€ = awsay, 
QqN =x, QCN = yx ,CQq=x-x’ =y¥ say 
we have exactly, from - geometry of the ellipse, 
tan ff = -_ tan £ (r) 
and an identical expression in y. Let 
p= a — Pe ee £ -3S +15, etc... (8) 
then from Lagrange’s development we may obtain 
a= {= — psin 2£-hp? sin 4f-ete.........- (t) 
1 This can in no case involve an error of 0°”005 as previously pointed out. 
