May 24, 1883] 



NATURE 



87 



water, diffuse into one another ; the intervening layer 

 must have a practically " stationary " refractive index at 

 each of its bounding surfaces, and a stratum of greatest 

 rate of change of index about midway between them. 

 The exact law of change in the stratum is a matter of 

 comparatively little consequence. I have assumed (after 

 several trials) a simple harmonic law for the change of 

 the square of the refractive index within the stratum. 

 This satisfies all the above conditions, and thus cannot 

 in any case be very far from the truth. But its special 

 merit, and for my purpose this was invaluable, is that it 

 leads to results which involve expressions easily calcu- 

 lated numerically by means of Legendre's Tables of 

 Elliptic Integrals. This numerical work can be done once 

 for all, and then we can introduce at leisure the most 

 probable hypotheses as to the thickness of the transition 

 stratum, the height of its lower surface above the 

 ground, and the whole change of temperature in passing 

 through it. I need not now give the details for more 

 than one case, and I shall therefore select that of a tran- 

 sition stratum 50 feet thick, and commencing 50 feet above 

 the ground. From the physical properties of air, and the 

 observed fact that the utmost angular elevation of the 

 observed images is not much more than a quarter of a 

 degree, we find that the upper uniform layer of air must 

 under the conditions assigned be about 7 C. warmer than 

 the lower. Hence by the assumed law in the stratum, 

 the maximum rise of temperature per foot of ascent (about 

 the middle of the transition stratum) must be about o°'2 C. 

 per foot. Such changes have actually been observed by 

 Glaisher in his balloon ascents, so that thus far the hypo- 

 thesis is justified. But we have an independent means 

 of testing it. The form of the curve of vertices is now 

 somewhat like the full lines in the following cut, Fig. 5 : — 



Fig. 5. 



where E is the eye, and the dotted lines represent the 

 boundaries of the transition stratum. It is clear that, if 

 PM be the vertical tangent, there can be but one image 

 of an object unless its distance from E is at least twice 

 E M. This will therefore be called the " Critical dis- 

 tance." If the distance be greater than this there are 

 three images : — one erect, seen directly through the lower 

 uniform stratum- -then an inverted one, due to the diminu- 

 tion of refractive index above the lower boundary of the 

 transition stratum — and finally an erect image, due to 

 the approximation to a stationary state towards the upper 

 boundary of that stratum. Now calculation from our 

 assumed data gives E M about six miles, so that the 

 nearest objects affected should be about twelve miles off. 

 Scoresby says that the usual distance was from ten to 

 fifteen miles. Thus the hypothesis passes, with credit, 

 this indep_ndent and severe test. A slight reduction of 

 the assumed thickness of the transition-stratum, or of 

 its height above the ground, would make the agreement 

 exact. 



All the phenomena described in Vince's paper of 1799, 

 as well as a great many of those figured in Scoresby's 

 works, can easily be explained by the above assumptions. 

 Scoresby's remarkable observation of a single inverted 

 image of his father's ship (when thirty miles off, and of 

 course far below the horizon) requires merely a more 

 rapid diminution of density at a definite height above the 

 sea. His figure is the second in Fig. 1 above. But 

 Scoresby figures, as above shown, several cases in which two 

 or more inverted images, without corresponding erect ones, 

 were seen abovethe ordinarydirect image. The natural ex- 



planation is, of course, a series of horizontal layers of up- 

 ward diminishing density and without a " stationary state " 

 towards their upper bounding planes. I find that, by 

 roughly stirring (for a very short time) a trough in 

 which weak brine below is diffusing into pure water 

 above, we can reproduce this phenomenon with great 

 ease. In fact, when temporary equilibrium sets in, the 

 fluids are arranged in a number of successive parallel 

 strata with somewhat abrupt changes of density. 



But the mathematical investigation, already spoken of, 

 shows that it is quite possible that there may be layers 

 tending to a stationary state without any corresponding 

 visible images. 



This depends on the fact that, while the inverted image 

 (due to the lower part of a stratum) is always taller than 

 the object seen directly (though not much taller unless 

 the object is about the critical distance) ; the numerical 

 calculation shows that the erect image is in general ex- 

 tremely small, and can come into notice only when the 

 object is not far beyond the critical distance. Thus there 

 may have been, in all of Scoresby's observations (though he 

 has only occasionally noticed and depicted them) an erect 

 image above each inverted one, but so much reduced in 

 vertical height as to have been invisible in his telescope, 

 or at least to have formed a mere horizontal line so 

 narrow that it did not attract his attention. The greatly 

 superior number of inverted images, compared with that 

 of the direct ones, figured by Scoresby may thus be 

 looked upon as another independent confirmation of the 

 approximate correctness of the hypothetical arrangement 

 we have been considering. 



To obtain an experimental repetition of the phenomena 

 in the manner indicated by the above hypothesis, a tank, 

 with parallel glass ends, and about 4 feet long, was half- 

 filled with weak brine (carefully filtered). Pure water 

 was then cautiously introduced above it, till the tank was 

 nearly filled. After a few hours the whole had settled 

 down into a state of slow and stealy diffusion, and 

 Vince's phenomenon was beautifully shown. The object 

 was a metal plate with a small hole in it, and a lamp with 

 a porcelain globe was placed behind it. I he hole was 

 triangular, with one side horizontal (to allow of distinction 

 between direct and inverted images), and was placed near 

 one end of the tank, a little below the surface-level of the 

 unaltered brine, the eye being in a corresponding position 

 at the other end. A little vertical adjustment of object 

 and eye was required from time to time as the diffusion 

 progressed. The theoretical results that the upper erect 

 image is usually much less than the object, and that it is 

 seen by slowly convergent rays, while the inverted image 

 is larger than the object and is seen by diverging rays, 

 were easily verified. 



To contrast Wollaston's best-known experiment with 

 this, a narrow tank with parallel sides was half-filled with 

 very strong brine, and then cautiously filled up with pure 

 water. (The strong brine was employed to make up, as 

 far as possible, for the shortened path of the rays in the 

 transition stratum.) Phenomena somewhat resembling 

 the former were now seen, when object and eye were 

 nearly at the same distance apart as before, and the tank 

 about half-way between them. But in this case the dis- 

 parity of size between the images was not so marked — 

 the upper erect image was always seen by diverging rays, 

 the inverted image by rays diverging or converging 

 according as the eye was withdrawn from, or made to 

 approach, the tank. In this case, the curvature of each 

 of the rays in the vessel is practically constant, but is 

 greatest for the rays which pass most nearly through the 

 stratum of most rapid change of refractive index. Hence, 

 when a parallel beam of light fell horizontally on the tank 

 and was received on a sufficiently distant screen, the 

 lower boundary of the illuminated space was blue — and 

 the progress of the diffusion could be watched with great 

 precision by the gradual displacement of this blue band 



