552 PROFESSOR TAIT ON MIRAGE. 
VincE, and also in a few cases by Scoresby, involves an inverted image with a 
direct image above it. In some other cases observed by Scorgssy, the direct 
or the inverted image alone was seen, the object itself being situated far below the 
horizon. Some excerpts from Scoressy’s figures (which are themselves com- 
posite) are given in fig. 1. A comparison of these observations with VINCE’s 
diagram of the supposed courses of the rays seemed to me to show that a single 
transition stratum may be capable of giving either a single image, direct or in- 
verted according to circumstances, or an inverted image with a direct image 
above it. As, in at least the greater number of the observations to which I 
have referred, both the object and the spectator seem to have been below the 
transition stratum which caused the phenomena, I do not think that Wot- 
LASTON’s square bottle with two inter-diffusing liquids presents a fair analogy. 
For, with that arrangement, the rays enter and emerge from the transition 
stratum by its ends, and not by its lower side, as, from VINcE’s diagram, they 
would appear to do in nature. 
I propose to return to the consideration of this arrangement of WoLLASTON’s. 
But meanwhile I will sketch (1) the mode in which I was led to see that, under 
proper conditions, a simple continuous law of refractive index may lead to the 
formation of three images, (2) how the consideration of the mode in which these 
are produced-in a medium whose refractive index varies to four-fold or more of 
the minimum value, led me by necessary steps to see how they can be produced 
in the lower atmosphere whose refractive index can vary, even in extreme cases, 
by only zo too OF SO. 
2. To fix the ideas, we will begin with a particular case, which is a thoroughly 
illustrative one so far as theory is concerned, and is also interesting as it 
reproduces, with singular accuracy, the exaggerated diagram by which VINCE 
endeavoured to explain his observations. 
The ordinary characteristic of a maximum or minimum is that it differs from 
neighbouring values of the function by a quantity depending on the square of 
the increment of the independent variable. Assuming then, without any 
inquiry as to the other physical circumstances, the existence of a medium 
whose refractive index is represented by the equation 
w=a+y? : 
it is clear that y=0 is a plane of minimum refractive index. 
HAmILton’s equation for this case is, 7 being the characteristic function, 
ary GN ieee at “alle 
since it is obvious that the path is in a plane perpendicular to y=0. 
