258 Prof. Everett on the Optics of Mirage. 



inclination of ray is the same in all parts of the medium, 

 have 



dx a 



We 



or, putting e h v=iZ } 



dx a 1 1 ...... * fll 





dz b zVz 2 — a 2i b a 





Making c=0, this gives e b ^ = asecbx ; and y will be infinite 



rjr rrr- 



when bx= — so that the line #== ht ^ s an asymptote. 



Some, at least, of the foregoing seven cases are to be found in 

 existing text-books and examination-papers ; but I believe that 

 the method here set forth is an improvement upon that generally 

 adopted. 



XVIII. The following additional proof of the general law of 

 ray-curvature is interesting, as being directly deduced from the 

 law of least time. 



Let T denote the time of passage of a ray from a fixed point 

 to the point whose coordinates are x, y, z. Let a, /3, 7 be the 

 angles which the forward direction of the ray at x, y, z makes 

 with the axes, and let /jl be the index and v the velocity at x, y, z, 



y 

 so that fi= — , where V denotes the velocity in vacuo. 



Draw a short line from x, y, z in any direction perpendicular 

 to the ray at this point, and let its direction-cosines be /, m, n. 

 By the principle of least time (or, more correctly, of stationary 

 time), the value of T will be the same (to the first order of small 

 Quantities) from the fixed point to both ends of this short line ; 

 that is, 



But also 

 therefore 



dx dy dz 



I cos u + m cos /3 -f n cos 7=0; 



h suppose. 



^T ^T ^T 



dx dy dz 



cos « cos/3 cos 7 



Again, if we draw a short line along the ray from x, ij, z, tne 

 values of T from the fixed point to the two ends of this line will 

 differ by the time of traversing this line. Hence we have 



rfT dT k. dT 1 



-7- COS d + -r-CO$B+ -j— COS 7 ss - • 



ax dy dz ' v 



