TERRESTRIAL REFRACTION AND MIRAGE. 271 
another point, horizontally distant from it, it takes a curved path as 
Fie. 2. 
a 
shown, much exaggerated, in the figure. The reason is this: suppos- 
inga ray of light to pass ina slightly oblique direction from this point, 
it 1s passing from dense air to air which is less dense, and accordingly 
the ray is bent away from the normal up to a certain point, and then on 
the downward path when the passage is from a less dense to a more dense 
air the ray is bent towards the normal; and that brings it down as shown. 
The ray does not take a path similar to that of a straight line bent intoa 
number of angles, but it takes a curvilinear path corresponding io the 
continuous change in the density of the medium. It may also be viewed 
in another manner by having regard to the velocity of light, and to the 
circumstance that light usually travels in the path that it can accomp- 
lish in the least time. In certain cases light travels in the path that it 
would take the longest time to traverse, 7.e. in the maximum time path ; 
but those cases do not occur in this particular portion of the subject. 
In terrestrial refraction light travels in the minimum path, and time is 
saved in passing from one point to another by curving up into 
strata of air through which it can pass with greater velocity—the 
velocity of air being greater in the less dense medium. Accordingly 
the time of description of this curvilinear path may be and is in fact less 
than the time that it would take to accomplish the straight path. It 
will be observed in this case that the path of the ray is curved with the 
concavity of the ray towards the denser layers, and that is a general 
principle in this subject. When a ray takes a curvilinear path the 
concavity is always presented towards the denser side. There is also 
another principle which guides us, and that is that the amount of cur- 
vature of the ray, as measured by the change of direction for a given 
length of ray, is directly proportional to the rate at which the density 
of the air changes along the normal. or instance supposing a vertical 
ray coming down through the strata, the normal is horizontal and there 
is no change of density horizontally ; accordingly there will be no cur- 
vature, and the ray will pass in a straight line. But if we take a ray, 
the general direction of which is horizontal, the normal is very nearly 
vertical, and that is the direction alone which there is the maximum 
rate of change of density ; and accordingly we get the maximum degree 
of curvature. We get the minimum curvature for vertical rays, the 
maximum curvature for horizontal rays, and intermediate amounts of 
curvature for oblique rays. 
The way that the refraction is found is as follows: supposing Fig. 3 
to represent the earth, and A and B to represent two places on the sur- 
face of the earth, and 4 B a straight line joining them, then A P B will 
