PROFESSOR TAIT ON MIRAGE. 559 
The general problem of determining the images is, in this case, a very 
complicated, though not difficult, one; but it becomes much simplified if we 
assume as before the object and eye to be at the same level. It is obvious 
that a vertical line, midway between the eye and the object, will cut the curve 
_of vertices an infinite number of times, both above and below the maximum 
stratum. Thus there is in such a case an infinite number of images, which are 
seen by rays which have crossed the maximum stratum an even number of 
times, in which zero may be included. These must each have one, or some 
other odd number, of vertices between the eye and the object, and the hori- 
zontal distance between two such vertices is 
7 Va—7? ; 
which is therefore less for that one of two rays which intersects the maximum 
plane at the greater angle. . 
In nature, of course, the number of images depending on a law like this 
must always be finite, because the utmost percentage change of refractive index 
in the lower atmosphere is very small. But, independent of equilibrium 
considerations, there is the farther objection that it cannot be reconciled with 
the appearances seen by VINCE and Scoressy. For these were, in the main, 
very similar to one another for all distances of the object beyond certain 
limits ; while with the present assumption, the appearances presented by an 
object moving to successively greater distances would exhibit a species of 
guast periodic change which I have nowhere seen described. And, if we keep 
to probable changes in the refractive index of the atmosphere, this law will give 
only one image :—not, of course, in the true direction of the object :—but 
erect, and therefore not properly coming under the designation of “mirage.” 
10. After trying a number of assumptions as to the law of refractive index in 
the transition stratum, I finally chose for detailed examination the following :— 
pe=a?-+ecos ; 
This seemed to me particularly worthy of investigation, for it must be at 
least a fair approximation to the state of matters near the common boundary 
of two inter-diffusing fluids, or of two masses of the same fluid at different 
temperatures. This follows from the facts that :—it gives a stationary state at 
y=0, with a maximum refractive index ; and another at y=0, with a minimum 
index. Near y= = there is a stratum of greatest rapidity of change of index. 
This hypothesis has also the advantage of leading to equations which can be 
treated by the ordinary elliptic integrals. 
VOL. XXX. PART I. BES 
