PROFESSOR TAIT ON MIRAGE. 557. 
least =3'68a. Hence the refractive index at the level of the eye (Wa’+2’) 
must be at least 3°8 times that in the minimum stratum. And the distance at 
which an object on the horizon requires to be situated, in order that there may 
be three images of it, lies within exceedingly narrow limits, unless the 
refractive index at the level of the eye very greatly exceed this lowest admissible 
value. 
7. The possibility of three images of an object at the level of the eye evidently 
depends on the existence of three values of y, for the same value of in the 
curve of vertices. It is therefore necessary that we should study the question 
from this point of view. 
On thinking of the relative forms of the curves of vertices in fig. 2; the 
first of which gives only one image, the second and third (in certain cases) 
three :—I saw that the point of inflexion, on which the triple value of y 
depends, is due to the gradual diminution of curvature of the ray near the eye 
(for rays of a given inclination to the vertical) as the eye is placed lower in the 
medium. Hence any arrangement which lessens the curvature of the lower 
parts of the rays will increase this effect. 
In fact, the portion ABC of the ray OB (fig. 4) is congruent with the ray 
abe, if only the tangents at A and a be parallel. Hence the point B would 
be shifted to 6 if the ray Oa were straight (or at all events, less curved than 
OA) and the angle at a equal to that at A. 
Thus it was at once obvious that the curve of vertices (fig. 5) in the 
stratum above RS, might be made asymptotic to that line towards the right 
of the figure (the eye being still at O), if only the stratum below it were 
of uniform refractive index, or at least of a refractive index diminishing 
so slowly with increased height that a ray from O could intersect RS at a 
practically infinite distance. This at once showed me the general nature of 
one mode of explanation. The curve of vertices Q@PQ’ in the stratum RU will 
now be asymptotic, towards the right, to both RS and TU, and therefore can 
be cut in two points by a sufficiently distant vertical. These points correspond 
to VINCE’s two upper images, the third and lowest is seen by rays which have 
not reached the upper stratum, and for which the corresponding branch of the 
curve of vertices is the horizontal line OM, passing through the eye. 
8. To repeat :—the conditions requisite for the production of Vincr’s pheno- 
menon, at least in the way conjectured by him, are, a stratum in which the 
refractive index diminishes upwards to a minimum (or, at all events, nearly to 
a stationary state) ; and, below it, a stratum in which the upward diminution 
is either considerably less or vanishes altogether. The former condition (the 
