PROFESSOR TAIT ON MIRAGE. 555 
while (2) would have been (for rays passing through the point 0, d), 
0=1—-a uf Vig)—@ . . ° : . (2’) . 
The new form of (2’) shows that, for a given value of y, x increases with 
increase of a; provided no vertex is reached. For the denominator of the 
differential is less, and the integral is multiplied by a greater factor, than 
before. Hence two contiguous rays from the same point cannot again 
intersect till one, at least, has passed its vertex. When the vertex is included 
within the limits of integration, (2’) may by the symmetry of the ray be 
written 
aA == a? ste Tes ee noha =etaf” Tes Sar 
Now the middle term (as we have seen) is positive, and increases with a, 
if y>b. Hence the second intersection of the rays which have the common 
point 0, b, is at a point where y> 48, if and only if, € increases as a increases ; 
é., if the line, drawn from the vertex of the ray nearer to the minimum 
plane to that of the other, leans back towards the first common point of the 
two rays. The converse is easily seen to hold, by taking the second point of 
intersection as the starting-point and reversing the rays. Hence, if the 
minimum stratum be horizontal, two neighbouring rays, issuing from a common 
point below it, and originally directed above the horizon, intersect again before 
they have got back to the level of their former intersection, if their vertices 
be at a part of the curve of vertices where the tangent leans backwards over 
the starting-point, and vice versd. This proposition is, in fact, obvious 
from a mere inspection of the diagram fig. 3, in which the dotted curve is that 
of vertices, the eye being at E. 
To apply it to the case of phenomena such as those observed by VINCE and 
ScoRESBY, suppose the strata of equal refractive index to be horizontal. Then 
two rays slightly inclined to one another, leaving any point in a common 
vertical plane, will in general intersect one another before they again reach the 
level of the starting-point, if, and not unless, the vertex of the higher ray be 
horizontally nearer to the starting-point than that of the lower ray; 2.¢., if the 
part of the curve of vertices concerned leans towards the starting-point. Also, 
as is well known, when two rays slightly inclined to one another, cross once 
between the eye and the object, the image formed is an inverted one. 
5. Hence the following graphical method for finding the numberand characters 
