168 Prof. Everett on the Optics of Mirage. 



To obtain the convergence which the problem requires, we must 

 have 



dx* a*' K } 



for the general equation of the curved path of a ray will then have 

 the same form as before, namely 



2/= j sin £z£ ; ( F) 



and it is upon this form that the convergence which the problem 

 requires alone depends. For the same reason, the consequences 

 deduced in (III.) will remain completely applicable. 



Equation (D) now denotes the curve obtained by making 

 equal algebraic additions to the curvatures at all points of a 

 curve of sines, or by bending uniformly a rectangular rod in the 

 plane of one of its faces which has a curve of sines drawn upon it. 



The required law of variation of index is 









dlogfj, 



y 



a* 



1 

 If 









dy 





At the 



point 



where 



a ray cuts th 

 dlogfi 

 dy 



e axis 



1 

 R* 



of a? 



we 



have 



The axis of x } therefore, does not lie in the surface of maximum 

 index, but in that surface-of-equal-index which possesses the 

 property that rays cutting it at a small inclination have, at the 

 points of section, a curvature equal to that of the earth. In 

 consequence of this property, rays can travel for any distance 

 along that surface-of-equal-index which contains the axis of x ; 

 whereas rays above it have greater curvature and bend down to 

 meet it, while on the other hand rays below it are less curved 

 (or may even be curved in the opposite direction), and it accord- 

 ingly bends down to meet them. 



To find the level of no curvature, or of maximum index, we 



must put — ^^ =0, an equation which gives 



a' 



*=- r 



The depth of the surface of maximum index below the axis of x 

 is therefore a third proportional to the earth's radius and the 

 parameter a. Rays above this level are curved in the same di- 

 rection as the earth ; those below it are curved in the opposite 



a? 

 direction. The value of w is about five feet when the value of 



