SCORESBY'S SKETCHES 323 



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 (Fig. i), I have simply selected a few of the more remark - 



m 



Fig. i. 



able portions which bear on the questions to be discussed. It is to be remarked 

 that the angular dimensions of these phenomena 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 because they must also have on the whole consider- 

 ably 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 

 investigation, we may treat the problem as if the earth were a plane. This simplifies 

 matters very considerably, so that definite numerical results are easily obtained ; and 

 there is no difficulty in afterwards introducing the (comparatively slight) corrrections 

 due to the earth's curvature. But these will not be further alluded 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 considering the 

 well-known problem of the paths of projectiles discharged from the same gun, with 

 the same speed but at different elevations of the piece. This corresponds, in the 

 optical problem, to the motion of light in a medium the square of whose refractive 

 index is proportional to the distance from a given horizontal plane. Instead, how- 

 ever, of 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 inter- 

 sect again above the horizon. In the optical problem this corresponds to an erect 



412 



