gan G. H. KNIBBS, 
The circumference of the ellipse therefore lies outside the sun’s 
refracted disc, on the upper limb, and inside of the lower. In Fig. 
2, the dotted line represents diagrammatically the real boundary- 
of the sun’s disc, the firm line its boundary on the assumption 
that its circumference is an ellipse. The maximum difference 
between the upper and lower difference is 0-02; by pushing the 
computation farther it would be seen that their ratio is almost 
exactly as the upper and lower vertical contractions. 
For their values in the directions of the radii vectores, these 
vertical differences between the sun’s disc and the circumference 
of the ellipse, must be multiplied by cos 6.1 In this way we 
empirically find that the difference between the polar codrdinates 
of the refracted disc and the ellipse of the same axes is very 
accurately* represented by the expression & sin? 20, in which &= 
0:0057 c. This deviation depends mainly upon the fact that the 
variation of the refraction is not absolutely linear, particularly at 
low altitudes. It becomes sensibly so, however, for the sun’s 
diameter, at as low an altitude as 10°; at which the compression 
has diminished to about one-third of what it is at 5°, vide Table 
VII, §11. A little close consideration will show that the varia- 
tion of this term is of the same order as (33), and that it rapidly 
becomes negligible with increase of altitude. It is convenient 
therefore, to combine it with that expression, by adding for the 
upper limb, and subtracting for the lower, 2 of the coéfiicient.* 
Hence it becomes $(1 + #), and the whole expression for the 
contraction of the reduced or contracted horizontal semidiameter, 
may be written 
1 The result though of course theoretically only approximate, is pet- 
fectly exact to the order of the quantities under consideration. We may 
even take = 223° etc. in the case considered. 
2 The mean results are -061, °130, ‘079; the expression gives ‘069, °135, 
“06. 
3 The value of $c?/S1 is for the mean value of ¢ for the upper and 
lower contracted semidiameters is 0-”223 and of k 0-135, that is very 
approximately 2 of that amount, 
