Prof. Everett on the Optics of Mirage. 251 



axis will converge to a geometrical focus also situated upou it, 

 as proved (for rays in one plane) in the previous paper. 



XII. The path of a ray under the conditions treated in last 

 section is the same as that of a particle attracted towards the 

 axis of x by a force varying directly as the distance. For, let 



v 



the intensity of force at distance r be -%; then the components 



v z 



parallel to the axes of y and z will be ^ an( ^ ~2> s0 tna t tne dif- 

 ferential equations of motion will be 



d?x _~ d 2 y _ y d 2 z _ z 

 3F ' dt*~~"a*' ~aW~~~tf' 



From the first of these equations we have x = kt-\-k' } or, reckon- 

 ing time from the instant when # = 0, and making the constant 

 ^-component of velocity unity, 



x = t. 

 Hence, putting x for t in the other two equations, we have 

 d^y _ y d*z _ z 

 dx q a 2 dx 2 a? 



which are identical with the equations to the ray. 



In like manner, the path of a ray under the conditions treated 

 in Section II. of the previous paper is the same as that of a par- 

 ticle attracted towards the plane of reference by a force varying 

 directly as the distance from this plane. 



The differential equations of the motion of the particle are not 

 (like those of the ray) limited to paths of small inclination. 



As regards velocity, that of the particle increases and that of 

 the ray diminishes as the line or plane of reference is approached. 



Thus far most of our reasoning respecting rays has been 

 merely approximate, and applicable only to rays of small incli- 

 nation to the surfaces-of-equal-index. We now proceed to some 

 rigorous deductions from the law of ray-curvature. 



XIII. Let the surfaces-of-equal-index be concentric spheres. 

 Required the law of index-variation which will cause all circles 

 described about the common centre to be paths of rays. 



Putting r for distance from centre, the required condition is 



that is, 

 or 



1 _ 1 _ 6? log fl . 

 r p dr 



dhgr + dhg/jb — O, 



/*r = C, an arbitrary constant. 



