Prof. Everett on the Optics of Mirage. 255 



of-equal-index being still parallel planes) which will cause all 

 rays to be arcs of circles. 



Let a and b.he the coordinates of the centre of one of these 

 circles, the ray being supposed to pass through the origin, and 

 the radius of the circle being called c. Then the equation of the 

 ray will be 



x <2 -2ax + y' 2 -2by = 0, 



„ ds .„ , c , . . i v 



; and since p is c, we have by equa- 



1 



tne vame 



or -y- Will 



dx 



e r 



y-b 



tion (L), 



dlos; p, 



1 ds 



that is, 



dy 



p dx 



or 



c y — b y — b } 



d\ogfi=—dhg{y-b), 



y — b clearly denotes distance from a fixed plane, parallel to the 

 plane of reference, and passing through the centre of the circular 

 ray considered. The result which we have obtained accordingly 

 indicates that, if the value of pu on one side of the plane of refer- 

 ence vary inversely as distance from a fixed plane on the other 

 side of and parallel to the plane of reference, all rays on the first- 

 mentioned side will be arcs of circles, having their centres in 

 the said fixed plane. 



If the medium is symmetrical about the plane of reference, 

 there will be two such planes of centres, and the complete course 

 of a ray will be a wavy line consisting of equal circular arcs suc- 

 ceeding one another in reversed positions. The half-wave- 

 length may have any value from zero to infinity, the expression 

 for it being 



2a = 2 V~d^¥ t 



where c varies from one ray to another, while b is constant. 



When the two planes of centres merge into one (or, in other 

 words, when b is zero), the arcs become semicircles, and a curious 

 question arises as to the course of a ray after cutting the plane 

 of reference at right angles. If a ray once become normal to 

 the planes-of-equal-index, what is to make it swerve to one side 

 more than to the other ? The difficulty vanishes, or at least is 

 indefinitely postponed, when we remember that the velocity of 

 the ray, being inversely as p,, diminishes to zero in approaching 

 the plane of reference, and infinite time will be required to reach 

 this plane. 



. The results which we have established in the present section 

 might have been deduced at once from the undulatory theory 



