of Edinburgh, Session 1881-82. 
355 
The main feature of this improvement is the study of the curvature 
of the ray in terms of the rate of change of refractive index in the 
medium. 
It seemed, however, to the author that a still simpler and, for 
some purposes, more powerful method could he made to depend 
upon the study of the curve on which lie the vertices of all rays 
passing through a point and confined to one plane perpendicular to 
the strata of equal refractive index. For the path of every ray is 
necessarily symmetrical about an axis perpendicular to these strata. 
Now suppose the strata to be horizontal, which is the common 
case, 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 ; i.e., 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. 
Hence the following simple graphical method for finding the 
number and characters of the images of an object situated at or 
near to the horizon. Draw the curve of vertices for all rays leaving 
the eye in the vertical plane containing the object. Draw also a 
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. 
It is easy to see that at these intersections the curve of vertices 
must alternately lean from, and towards, the eye, i.e., 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. 
Thus, as has long been known, the vertices of all the 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 from the gun, the upper half towards it, and these correspond 
2 z 
VOL. XI. 
