May 24. 1883J 



NA TURE 



85 



ciently comprehensive ; while some of his experiments 

 imitate closely the state of the air requisite for the pro- 

 duction of Yince's phenomena. Had Wollaston only felt 

 the necessary confidence in his own theory, he could 

 hardly have failed to recognise that what he produced by 

 the extreme rates of change of temperature in the small 

 air-space close to a red-hot bar of metal, could be pro- 

 duced by natural rates of change in some ten or twenty 

 miles of the atmosphere :— and he would have deserved 

 the credit of having completely solved the problem. 



Six months after my paper was read, another happy 

 chance led me to seek for a voluminous paper by Biot, of 

 which I had seen no mention whatever in any of the 

 books I had previously consulted. The probable reason 

 for the oblivion into which this treatise seems to have 

 fallen is a curious one. It forms a considerable part of 

 the volume for 1809 of the Mint- de I'Institut. But in 

 the three first great libraries which I consulted, I found 

 this volume to be devoid of plates. In all respects but 

 this, each of the sets of this valuable series appeared to 

 be complete. Without the figures, which amount to no 

 less than sixty-three, it is practically impossible to under- 

 stand the details of Biot's paper. The paper was, how- 

 ever, issued as a separate volume, " Rc'cherchcs stir les 

 Refractions extraordinaires qui out lieu pres de V horizon " 

 (Paris, 18101, which contains the plates, and which I 

 obtained at last from the Cambridge University Library. 

 I have since been able to procure a copy for the Edin- 

 burgh University Library. Biot's work is an almost 

 exhaustive one, and I found in it a great number of 

 the results which follow almost intuitively from my 

 methods : — such as the possible occurrence of four 

 images, under the conditions usually assumed for the 

 explanation of the ordinary mirage ; the effects of (un- 

 usual) refraction on the apparent form of the setting 

 sun ; &c. But it seems to me that Biot's long-continued 

 observations of the phenomena as produced over extensive 

 surfaces of level sand at Dunkirk have led him to take a 

 somewhat onesided vie* of the general question. And, 

 in particular, I think thn his attempted explanation of 

 Vince's observations (so far as I am able to understand 

 it; for it is very long, and in parts extremely obscure and 

 difficult, besides containing some singular physical errors) 

 is not satisfactory. His general treatment of the whole 

 question is based to a great extent upon the properties of 

 caustics, though he mentions (as the courbe des minima') 

 the "locus of vertices" which I had employed in my 

 investigations, and which I think greatly preferable. 

 There can be no doubt, however, that Biot's paper comes at 

 least next in point of importance to that of Wollaston : — 

 though in his opinion Wollaston' s work was complete 

 only on the physical side of the problem. "Sous le rap- 

 port de la physique son travail ne laisse rien a. desirer." 



But, if the chief theoretical papers on the subject have 

 thus strangely been allowed to drop out of notice, the 

 case is quite different with several of those which deal 

 with the observed phenomena. Scoresby's Greenland, 

 his Arctic Regions, and his Voyage to the Northern Whale 

 Fishery, are still standard works ; and in them, as well as 

 in vols. ix. and xi. of the Trans. R.S.E., he has given 

 numerous careful drawings of these most singular appear- 

 ances. The explanatory text is also peculiarly full and 

 clear, giving all that a careful observer could have been 

 expected to record. It is otherwise with the descriptions 

 and illustrations in Vince's paper {Phil. Trans. 1799). 

 In fact the latter are obviously not meant as drawings of 

 what was seen ; but as diagrams which exhibit merely 

 the general features, such as the relative position and 

 magnitude of the 'mages : — the details being filled in at 

 the option of the engraver. That such was the view 

 taken by Brewster, is obvious from the illustrations in his 

 Optics (Library of Useful Knowledge), for while one of 

 Scoresby's drawings is there copied, one of Vince's is 

 treated in a highly imaginative style by the reproducer. 



Scoresby's sketches are composite, as he takes care to 

 tell the reader, so that in the reproduction below [Tig. 1) I 

 have simply selected a few of the more remarkable portions 

 which bear on the questions to be discussed. It is to be 



Fig. i. 



remarked that the angular dimensions o c these pheno- 

 mena are always of telescopic magnitude : — the utmost 

 elevation of an image rarely exceeding a quarter or a 

 third of a degree. 



Because the rays concerned are all so nearly horizontal, 

 and (on the whole) concave towards the earth ; and be- 

 cause they must also have on the whole considerably 

 greater curvature than the corresponding part of the 

 earth's surface, especially if they happen to have points 

 of contrary flexure ; it is clear that, for a preliminary in- 

 vestigation, we may treat the problem as if the earth were 

 a plane. This simplifies matters very coasideribly, so 

 that definite numerical results are easily obtained ; and 

 there is no difficulty in afterwards introducing the (com- 

 paratively slight) corrections due to the earth's curvature. 

 Hut these will not be farther allufed to here. 



Of course I began, as almost every other person who 

 has thought of the production of the ordinary mirage of 

 the desert must naturally have begun, by co isidering the 

 well-known problem of the paths of projectiles discharge! 

 from the same gun, with the same speed but at different 

 elev.itions of the piece. This corresponds, in the optical 

 problem, to the motion of light in a medium the square 

 of wh ise refractive i.idex is proportional to the distance 

 from a given horizontal plane. Instead, ho.vever, ot 

 thinking chiefly of the different elevations corresponding 

 to a given range, I sought for a simple criterion which 

 should enable me to decide (in the optical application) 

 whether the image formed would, in any particular case, be 

 a direct or an inverted one. And this, I saw at once, could 

 be obtained, along with the number and positions of the 

 images, by a study of the form of the locus on which lie 

 the vertices of all the rays issuing from a given point. 

 Thus, in the ballistic problem, the locus of the vertices of 

 all the paths from a given point, with different elevations 

 but in the same vertical plane, is an ellipse. 



Its minor axis is vertical, the lower end being at the 

 gun ; and the major axis (which is twice as long) is in the 

 plane of projection. Now, while the inclination of the 

 piece to the horizon is less than 45 , the vertex of the path 

 is in the lower half of this ellipse, where the tangent 

 leans forward from the gun ; and in this case a small 

 increase of elevation lengthens the range, so that the two 

 paths do not intersect again above the horizon. In the 

 optical probleai this corresponds to an erect image. ^ But, 

 when the elevation of the piece is greater than 45 , the 

 vertex of the path lies in the upper half of the ellipse, 

 where the tangent leans back over the gun ; and a small 

 increase of elevation shortens the range. Two contiguous 

 paths, therefore, intersect one another again above the 

 horizon. And, in the optical problem, this corresponds 

 to an inverted image. In symbols, if the eye be taken as 

 origin and the axis of x horizontal, there will be a direct 

 image for a ray at whose vertex dy/dx and x (in the curve 

 of vertices) have the same sign, an inverted image when 

 the signs are different. 



