of Edinburgh, Session 1881-82. 
357 
curve can be cut in three points by a vertical line. Thus it was 
seen that the triple images of ships in the offing, described by Yince 
in the Baker ian Lecture for 1799, can be explained by the assump- 
tion of one plane of minimum index ; or rather, as the effective 
rays never reach this plane, one plane of (approximately) stationary 
index. But this particular assumed law leads to variations of re- 
fractive index far more rapid than can possibly exist in still air, and 
shows the two upper images (where they exist) at a very great 
elevation. 
Other assumptions were then made, such as 
Ja^ + if 
These forms were assumed so as to give a stationary plane, and 
also to give expressions (in the equation of the curve of vertices) 
integrable by circular, logarithmic, or elliptic functions. They all 
gave results more or less resembling that already described, showing 
three images, but giving changes of refractive index not likely to 
occur in nature. 
The author then tried the supposition of a transition stratum of 
finite thickness, in which the refractive index should vary continu- 
ously from that of the uniform denser fluid below to that of the 
uniform rarer fluid above. This is obviously the sort of arrange- 
ment which is most likely to occur in nature. Also the calculations 
already made for the former assumptions could be utilised for this 
by assuming the former laws of variation of refractive index to hold 
in the transition stratum only. Then it was at once evident that the 
curve of vertices in the transition stratum is asymptotic to both 
boundaries of the stratum, its vertex being turned towards the 
eye, provided the upper boundary be a stationary one, and it 
therefore admits of being cut in two points by any vertical line 
which is distant from the eye by more than a certain calculable 
amount. This arrangement is easily shown to account completely 
for the two upper images seen by Yince. The lower one is, of course, 
