CURVE OF VERTICES 



325 



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 curvatures 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 



Fig. 3- 



would be of the form in Fig. 3 (the drawing is, of course, immensely exaggerated) 

 and the locus of vertices, A, B, C, obviously leans from the eye. But now suppose 

 that, below a stratum of this kind, there were one of constant density, in which of 



Fig. 4. 



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 the upper medium being 



