330 G. H. KNIBBS. 
tion P P’ by c,; the angle of the radius vector C P, i.¢., the angle 
MCP by @; and the intercept P Q’ between the two arcs, of a 
line parallel to C M by c,, then we have by geometry 
8,/(S, cos 6) = e/e,; orc, = (S,¢ cos 0)/8,........ (p) 
and p Q’ being at right angles to C P 
¢, = Pp + pP’ =c¢, cos +(e} sin? 0)/(25 —ete.)...(¢) 
with a high order of precision! By a method of successive 
approximations the following values of c, and c, may be derived, 
viz., 
¢, = ccos O[1 + © sin? 0 (1 — 3 © cos?9)] (31) 
S, Ss. a 
eae 3 Cc : c 2 
Cs =c¢ Oe re ano [bea (1 + cos? )]}....... (32) 
The second term, of this value for the contraction of the inclined 
semidiameter, is generally negligible. It may be reéxpressed thus 
for the purposes of calculation,? 
= oS sin? 26 (33) 
3c? 
28, 
and is therefore obviously a maximum for 6= 45°. Resuming 
the previous example, in which the true altitude of the sun’s 
centre is taken as 5°, and the difference of the contractions con- 
sequently 22-’69, we have S, = 15’ 37-"01, hence the value of the 
term is 0°21. This quantity is of the same order as the difference, 
referred to in the preceding section, between the real image by 
refraction, and that deduced from the assumption of elliptical 
form. If the second term therefore be regarded as appreciable, 
the defect of the elliptical hypothesis must be considered at the 
same time. 
In order to illustrate the difference between the elliptical outline 
and the figure given by the refraction theory, let us revert to the 
case where the sun’s centre has a true zenith distance of 85°, its 
apparent zenith distance being therefore 84° 50’ 28-36 at the 
The more complete expression for the re of the = term 
in (q) is 2 S—(c? sin? §)/2 8 - ete., a continued fra 
? The computation of the squares of sines and cosines is facilitated by 
using the formule sin* a = } (1-cos 2a) and cos? a = } (1-+cos 2a). 
