86 



NATURE 



[May 24, 1883. 



Hence, whatever be the law of refractive index of the 

 air, provided only it be the same at the same distance 

 from the earth's surface, (i.e. the surfaces of equal density 

 parallel planes, and therefore the rays each symmetrical 

 about a vertical axis) all we have to do, in order to find 

 the various possible images of an object at the same level 

 as the eye, is to draw the curve of vertices for all rays 

 passing through the eye, in the vertical plane containing 

 the eye and the object, and find its intersections with the 

 vertical line midway between the eye and the object. As 

 soon as this simple idea occurred to me, I saw that it 

 was the very kernel of the matter, and that all the rest 

 would be mere detail of calculation from particular hypo- 

 theses. Each of the intersections in question is the 

 vertex of a ray by which the object can be seen, and the 

 corresponding image will be erect or inverted, according 

 as the curve of vertices leans from or towards the eye at 

 the intersection. Thus, in Fig. 2, let E be the eye, and 



Fig. 2. 



the dotted line the curve of vertices for all rays in the 

 plane of the paper, and passing through E. Let A be an 

 object at the level of the eye, A 1 A 2 A 3 the vertical line 

 midway between E and A. Then A 1 , A 2 , A 3 are the 

 vertices of the various rays by which A can be seen. If 

 we make the same construction for a point B, near to a, 

 we find that whereas the contiguous rays through A 1 , B 1 

 and through A 3 , B :i do not intersect, those through A'-, e- 

 do intersect. At A 1 and A 3 the curve of vertices leans 

 from the eye, and we have erect images ; at A 2 it leans 

 back towards the eye, and we have an inverted image. 

 And thus, if this curve be continuous, the images will be 

 alternately erect and inverted. The sketch above is 

 essentially the same as one given by Vince, only that he 

 does not employ the curve of vertices. If the object and 

 eye be not at the same level, the construction is not quite 

 so simple. We must now draw a curve of vertices for 

 rays passing through the eye, and another for rays 

 passing through the object. Their intersections give all 

 the possible vertices. (This construction of course gives 

 the same result as the former, when object and eye are at 

 the same level.) But the images are now by no means 

 necessarily alternately erect and inverted, even though 

 the curve of vertices be continuous. However, I merely 

 note this extension of the rule, as we shall not require it 

 in what follows. 



I then investigated the form of the curve of vertices in 

 a medium in which the square of the refractive index 

 increases by a quantity proportional to the square of the 

 distance from a plane in which it is a minimum, and 

 found that (under special circumstances, not however 

 possible in air) three images could be produced in such a 

 medium. But the study of this case (which I could not 

 easily explain here without the aid of mathematics) led 

 me on as follows. 



As the curvature of a ray is given by the ratio of the 



rate of change of index per unit of length perpendicular 

 to the ray, to the index itself (a result which I find was at 

 least virtually enunciated by Wollaston) ; and as all the 

 rays producing the phenomena in question are very nearly 

 horizontal: — i.e. perpendicular to the direction in which 

 the refractive index changes most rapidly: — their curva- 

 tures are all practically the same at the same level. 

 Hence if the rate of diminution of the refractive index, 

 per foot of ascent, were nearly constant, through the part 

 of the atmosphere in which the rays travel, the rays we 

 need consider would all be approximately arcs of equal 

 circles ; and the curve of vertices would (so far as these 

 rays are concerned) lean wholly from the eye ; being, in, 

 fact, the inferior part of another equal circle which has its 

 lowest point at the eye. Hence but one image, an erect 

 one, would be formed ; but it would be seen elevated 

 above the true direction of the object. This is practically 

 the ordinary horizontal refraction, so far as terrestrial 

 objects on the horizon are concerned. The paths of the 

 various rays would be of the form in Fig. 3 (the drawing 



Fig. \ 



is, of course, immensely exaggerated) and the locus of 

 vertices, ABC, obviously leans from the eye. But now 

 suppose that, below a stratum of this kind, there were one 

 of constant density, in which of course the rays would be 

 straight lines. Then our sketch takes the form Fig. 4 

 (again exaggerated) ; each of the portions of the ray in 



Fig. 4. 



the upper medium being congruent to the corresponding 

 one in the former figure (when the two figures are drawn 

 to the same scale), but pushed farther to the right as 

 its extremities are less inclined to the horizon. In its 

 new form the curve of vertices ABC leans back towards 

 the eye, and we have an inverted image. The lower 

 medium need not be uniform as, for simplicity, we 

 assumed above. All that is required is that the rate of 

 diminution of density upwards shall be less in it than in 

 the upper medium. 



Those who have followed me so far will at once see 

 that, as a more rapid decrease of density, commencing at 

 a certain elevation, makes the curve of vertices lean 

 back, so a less rapid decrease (tending to a " stationary 

 state '') at a still higher elevation will make the curve of 

 vertices again lean forward from the eye. I need not 

 enlarge upon this. 



Thus to repeat : — the conditions requisite for the pro- 

 duction of Vince's phenomenon, at least in the way con- 

 jectured by him, are, a stratum in which the refractive 

 index diminishes upwards to a nearly stationary state, 

 and below it a stratum in which the upward diminution is 

 either less or vanishes altogether. The former condition 

 secures the upper erect image, the latter the inverted 

 image and the lower direct image. 



In my paper read to the Royal Society of Edinburgh I 

 have given the mathematical details following from the 

 above statement : and have made full calculations for the 

 effect of a transition stratum, such as must occur between 

 two uniform strata of air of which the upper has the 

 higher temperature. From Scoresby's remarks ;t appears 

 almost certain that something like this was the state of 

 affairs when the majority (at least) of his observations 

 were made. When two masses of the same fluid, at 

 different temperatures, rest in contact ; or when two 

 fluids of different refractive index, as brine and pure 



