254 Prof. Everett on the Optics of Mirage. 



But the equation of any ray through the origin of coordinates is 

 to be of the form 



x 



y — b 



sin — ; 

 a 



whence 





dy b x 

 -f- = —cos-* 

 ax a a 



d 2 y b . x 

 dx 2 a 2 a 



Equation (L) therefore gives 





• % 

 71 — 6 sin- 

 olog yu, a —y 



dy 2 2 2 x a 2 + b 2 —y 2 

 a 



But — -j- — is to be simply a function of y, and therefore is not 



to vary from one ray to another. Hence the expression a 2 +b 2 

 in the denominator of the expression last found must be con-' 

 stant, and may be denoted by c 2 . The equation may then be 

 written 



d\ogfi=±d\og(c 2 -y 2 ), 

 or 



^=7^-^ 



/jb Q denoting the value of fi in the plane of reference. Hence it 

 is clear that when /jl varies as Vc 2 —y 2 , all rays will be curves 

 of sines ; and if b denote the amplitude of one of these curves, 

 its half-wave-length will be 



7rfl = 7r V 'c 2 —b 2 , 



which diminishes as amplitude increases. 

 For small amplitudes we have 



•i 



-b 2 = c\Jl- ^ =<?(.!- g^-j nearly. 



Hence the limiting value of half-wave-length, or, in other words, 



the geometrical focal length, is ire ; and the aberration from this 



b 2 

 focal length is always negative, its value being — 7r~-« 



When b (and therefore also y) is small compared with a, the 



value of — -~- above obtained becomes sensibly equal to — ^, 



thus confirming our previous approximate results. 



XVI. Next let us seek a law of index-variation (the surfaces- 



