556 PROFESSOR TAIT ON MIRAGE. 
of the images of an object situated at the level of the eye. Trace the curve of 
vertices for all rays leaving the eye in the vertical plane containing the object. 
Draw also a vertical line midway between the eye and the object. The inter- 
sections of this line with the curve of vertices are the vertices of all the paths 
by which the object can be seen, when the eye is in the assigned position. Or, 
what comes to the same thing, but (unlike the simpler construction) admits 
of application to an object at any level, draw the curve of the vertices as 
before, and then draw another for an eye placed at the object. Their intersec- 
tions determine the vertices of the rays giving all possible images. 
It is easy to see that, at the intersections with the vertical line midway 
between eye and object, the curve of vertices, if continuous, must alternately 
lean from, and towards, the eye, 7.¢., the images seen are alternately erect and 
inverted; their number depends of course upon the form of the curve of 
vertices ; which, in its turn, depends not only upon the law of refractive index 
in terms of level, but also upon the position of the eye. [This alternation of 
images does not necessarily hold when eye and object are at different levels. | 
Thus, as has long been known, the vertices of all the coplanar paths in 
which a projectile, fired with a given velocity, can move, with different 
elevations of the piece, lie in an ellipse whose major axis (double the minor 
axis) is horizontal. The lower half of this ellipse leans fromthe gun, the upper 
half towards it, and these correspond to angles of elevation of the piece, 
respectively less and greater than 45°. In the former case (when the 
elevation is less than 45°), a slight increase of elevation increases the range on 
a horizontal plane, so that the new path is wholly above the old one; which, 
however, would intersect it wnder the horizon. In the latter case a slight 
increase of elevation shortens the range, so that the two paths must intersect 
before reaching the ground. 
6. Recurring to the imagined medium in which 
we=a2+y2, 
we see by fig. 3 the paths of the rays by which the three images of AB are 
seen by an eye placed at E. This figure, as already remarked, is (with the 
exception of the introduction of the curve of vertices) almost identical with that 
of Vince in the Phil. Trans. for 1799. 
But it is easy to see that, although this shows the possibility of three 
images in the relative positions observed by V1NcE, it is in no way capable of 
explaining his observation. For the existence of three images, in such a 
medium, requires (as I have found by an approximate method)* that 6 be at 
* When 2 =0, we have 1 +S =a ke 1 tog (= + ns ye 371). Plotting the curves whose or- 
dinates (in terms of 7) are ae by these v5 quantities, ie find that they touch when b = 3°68a. 
