Nov. 19, 1874' 



NA TURE 



51 



again, any number of times, and every time that they do 

 so they will undergo a reversal of curvature. The curva- 

 ture at the point of crossing will bs nil. The curves de- 

 scribed will be what are called " harmonic curves," or 

 " curves of sines," such as arc represented in Figs, i and 2 ; 

 subject to the restriction that wc have only to do with 

 rays which are so nearly horizontal that the cosines of 

 their inclinitions may be treated as unity. The distance 

 betwccii consecutive intersections will bi the same for all 



v?^^g^|ivv|^^ 



ground, while the air is excessively transparent to his rays 



— flatness of surface, eminently conducive to the main- 

 tenance of unstable equilibrium — and absence of wind — 



such arc the conditions under which this form of mirage 



appe ir5. On the other hand, if the decrease of tempera- 

 ture upwards is slower than usual, the ordinary downward 



bending of rays will be increised, and if any physical 



cause, such as warm winds commencing aloft, before they 



are felt at the eartli's surface, produces a rcversil of the 



ordinary distribution of temperature, so 



that there is an increase upwards, instead 



of a decrease, this change will favour the 



downward bending of rays, which will, 



accordingly, be exaggerated ; for the 



lower air, being not only under greater 



pressure, but being also colder than the 



upper air, will for a double reason be 



denser. 



Capt. Scoresby states that " the curious 



refractions of the atmosphere in the polar 



regions are most frequent on the com- 

 mencement or approach of easterly wmds," 



and he elsewhere states that easterly and 



southerly winds are mild. 



An increase of temperature upwards, 



at the rate of about one-sixteenth of a 



degree Frhr. per foot, would make the 



curvature of rays equal to that of the 



earth, so that a ray might encircle the 



globe. Any increase in the downward 



bending of rays increases the range of 

 vision, by enabling them to bend round 

 the horizon, which previously limited the 

 view. The visible effect is precisely the 

 same as if the convexity of the surface 

 of the earth were diminished. And not 

 only will objects which were previously 



beyond the horizon be brought into view, 

 but objects which we:e previously visible 

 near the horizon will become plainer, inas- 

 much as the rays by which they are seen 

 will not pass so close to the intervening 

 surface as before, but will traverse a 

 higher portion of the air, which is less 

 liable to be ob cured by impurities. 



Having now laid doA'n the first prin- 

 ciples, to which all effects of atmospiieric 

 refraction must be traced, we will proceed 

 to some mere particular applications. 



I have recently been considering the 

 question — what must be the law of density 

 (or, more strictly, of refractive index) in 

 a horizontally stratified atmosphere, in 

 order that images formed by mirage may 

 be perfectly sharp ? and some of the 

 diagrams placed before you will serve 

 to explain the results which I have ob- 

 tained. 



P'irst. — Neglect the curvature of the 

 earth, and suppose the surface of uni- 

 form index to be plane; then the law 

 required is as follows : — There roust be 

 a place of maximum index, at which Plate II. 



the rate of variation of index with 



height must be zero ; and as we ascend or descend the curves, and is eas'ly computed in terms of the con- 

 from this plane of reference the rate of variation of index stant which enters into the expression for the variation of 

 must contmually increase in direct proportion to the dis- index. Apencilof rays diverging in the same vertical plane 

 tance. The rate must also be the same at equal distances from a point in the plane of reference, will thus converge 

 above and below this plane of reference. The curvature accurately to another point in the plane, as represented in 

 of a horizontal, or nearly horizontal ray, will thus be Fig i. Such a pair of points may be called principal 

 simply proportional to distance from the plane of refer- conjugate foci. But this property of accurate convergence 

 ence, and the bending from either side will be towards is not confined to pencils proceeding from points in the 

 this plane. Rays may accordingly pierce this plane (which plane of reference. The same property attaches to pencils 

 is indicated by a dotted line in Figs. I and 2) again and diverging from any point whatever ; the conjugate focus 



