that is, 



or 



Prof. Everett on the Optics of Mirage. 253 



d log fi = - \d log (a 2 -f 7" 2 ) , 



In a medium in which this law prevails, every helix of step 27r#, 



coaxial with the surfaces-of-equal-index, will be a ray-path ; and, 



conversely, every helical ray in the medium will have the step %ira. 



v 

 In the immediate vicinity of the axis, -^ - 2 is sensibly equal 



v 

 to -a, and the law of index now under consideration becomes 

 ar 



equivalent to that discussed in Section XL 



The step will be zero, and the helices will become circles, if a 

 be zero, in which case the law of index becomes /^r = C, as in 

 Section XIII. 



In general, when the surfaces of equal index are circular cy- 

 linders having for their common axis a line of maximum index, 

 helical ray-paths will exist at all distances from the axis, and to 

 each distance there will correspond a different step. The step 

 for any distance r will in fact be determined by putting 



d\og fi r 



dr d*-\- r 2 



and multiplying the value of a thus found by 2tt. 



XV. We found in Section II. of the previous paper that, for 

 rays of small inclination to converge to foci in a medium in which 

 /ju is a function of the distance y from a plane of reference, the 

 law of index-variation must be 



</log/^ ___ y 

 dy d l 



and that the rays will be curves of sines. Let us now ex- 

 amine the consequences of supposing all rays (whatever their 

 inclinations) to be curves of sines, the surfaces-of- equal-index 

 being still supposed to be parallel planes. 



The curvature - is, by the general law of ray-curvature, equal 



r 



dlog/j, dx , . . , dhf fda;\ 3 . 



t0 — d^r s> and ls a,so e<iual t0 ~ s»Uv by g eometr y- 



Hence we have 



d*y 

 dhgfj, 1 ds dx 2 



dy p dx 



-d) <L) 



