256 Prof. Everett on the Optics of Mirage. 



without any application of analysis. For if \x is inversely as the 

 distance from a fixed plane, velocity is directly as this distance, 

 and the velocities at all points in a plane wave-front are there- 

 fore directly as their distances from the intersection of the wave- 

 front with the fixed plane. Hence the wave-front will swing 

 round the line of intersection like a door upon its hinges, and 

 each point in the wave- front will describe a circular arc, which 

 will be the path of a ray. 



XVII. Instead of employing the differential equation (L), we 

 might have employed its integral, which can be reduced to the 

 form 



yLtcos 6 = a, 



a being a quantity which is constant for any one ray, but which 

 varies from one ray to another. This result has been previously 

 established in Section IV., and is in fact merely a statement of 

 the law of sines as applied to a medium in which the surfaces-of- 

 equal-index are parallel planes. 



Again, since the curvature of a ray depends only on the varia- 

 tion of log//-, we shall have precisely the same paths with the 



law fjb=f(y) as with the law jul ocf(y), since — -~- will have the 



y 



same value in both cases. The only limitation to this remark 

 depends upon the circumstance that^u, cannot be less than unity. 

 A small constant factor applied to f(y) may therefore diminish 

 the range through which the law /jl cc/(y) can hold. 



To obtain a convenient formula for determining the path of 

 rays when fi is given as a function of y, observe that 



see 2 — ^ —JSy)± (TVn 



S6t U - Oacosfl) 8 " a 2 ' ' * W 



+(gy 



whence 



dx 



d v ^{/Wl 2 -« 2 }' 



(N) 



which is the general differential equation to the paths of rays in 

 the medium, a being a parameter which varies from ray to ray. 

 The following are some of the cases in which the equation can 

 be integrated. 



dx CL 



(1) fi ccy, -j- = — g a > the differential equation of a ca- 

 tenary, the axis of x being the line which is commonly employed 

 as axis of x in treating of the properties of the curve, and a being 

 the distance of the vertex of the curve from this axis. 



The cases fiozy + b and /Jb(xb—y are reducible to this by 

 substitution. 



