568 PROFESSOR TAIT ON MIRAGE. 
possibly Scoresby, from insufficient telescopic power, failed to see (or at least to 
recognise as part of the phenomenon) the upper erect image, when the object 
was much beyond the critical distance. The table shows the great rapidity 
with which its height diminishes as the object recedes. The disparity between 
the images depends of course upon the fact that we have assumed a law which 
places the plane of most rapid change in the middle of the stratum. This may 
often not be the case in nature. It might be useful to work out the whole 
again, assuming a law (for the transition stratum) which would place the plane 
of most rapid change considerably out of the middle of the stratum. But I 
cannot attempt this at present. The results of § 14 seem also to be in com- . 
plete accord with ScorgEsBy’s observations at Bridlington Quay, which are the 
only detailed ones I have met with in which the point of view was shifted to 
or from the transition stratum. 
21. For an approximate estimate of the effect of the earth’s curvature on 
these phenomena, let us suppose the same law of density as before ; but let the 
strata be now J/evel, 7.¢., spheres concentric with the earth. The path of a ray 
in the lower stratum will still be straight, but the angle at which it meets the 
transition stratum (9+ 7, suppose) will now be necessarily greater than its 
original inclination (@) to the horizon. See fig. 7. 
If R be the radius of the earth, we find to a sufficient approximation, 
(R+c)cosp—R=Rw, 
or 
Ca 
aa » . 
As @ cannot be negative, the greatest value of w is 
2c 1 
R~ 460 
nearly ; ¢ being 50 feet, as before. If we write - for this quantity, we have 
ites 
2p0= ’ 
ae py 
whence, by giving py the values 1, 0°9, 0°8, &c., we easily obtain the following 
table :— 
6 1 0+ 
0:0000 0:0022 
0:0002 0:0022 
0:0005 0:0022 
0:0008 0:0023 
0-0012 0:0025 
0:0016 0:0027 
0:0023 0:0032 
0:0033 0:0040 
0:0053 0:0057 
00110 00112 
