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Proceedings of the Boyal Society 
seen directly through the nearly uniform air below the transition 
stratum. When the upper boundary is not at least nearly a station- 
ary one, the upper erect image is not formed. The figure, then, if 
inverted, gives the explanation of the ordinary mirage as seen in the 
desert. 
The author has since seen Wollaston’s account of his successful 
reproduction of Vince’s phenomenon in the stratum of air immedi- 
ately below a hot poker. It is quite clear that the explanation above 
given applies at once to this experiment, as well as in part to the 
other device employed by Wollaston, viz., the formation of a tran- 
sition layer between two masses of liquid of different densities which 
gradually diffuse into one another. The circumstances are here, 
however, not identical with the atmospheric ones, unless the vessel 
containing the liquids is of considerable length ; so that both object 
and eye are below the intermediate stratum. Of course, although a 
transition stratum is spoken of as the main cause of the phenomenon, 
it must not be supposed that above and below that stratum there is 
no change of refractive index. Any law of refractive index which 
makes a comparatively rapid decrease (ending with a nearly station- 
ary state) through a limited stratum is sufficient to account for 
the phenomena observed. 
The equation of the curve of vertices becomes in the case of a 
transition stratum of thickness b , 
\//(&) -ffi) ^ ’ 
where c is the height of the lower boundary above the eye. 
The rest of the paper treats of the application of Hamilton’s 
methods, the modifications of the above results due to the earth’s 
curvature, the apparent size, distance, brightness, &c., of the various 
images, and refers to other forms of the mirage phenomenon as 
observed, for instance, in distortions of the disc of the setting sun. 
Professor Everett, to whom the author had communicated some 
of the above results, had favoured him with the following letter 
(dated Dec. 3, 1881), expressing a desire that it should be read to 
the Society along with the author’s paper : — 
“ The following seems to me the best way of treating the medium 
