Prof. Everett on the Optics of Mirage. 163 



inclined to the surfaces-of-equal-index have sensibly the same 

 curvature at the same point. 



II. I now proceed to the solution of the following problem: — 

 Suppose a medium in which the index /n is a function of dis- 

 tance from a certain plane of reference which is itself a plane of 

 maximum index. It is clear from (I.) that rays cutting this 

 plane at any angle except a right angle will be bent back towards 

 it, and may, under proper conditions, meet it again. Required 

 the condition that rays cutting it at any small angle shall meet 

 it again at a constant distance — that is to say, at a distance sen- 

 sibly independent of 6. This is obviously the condition that 

 rays of small inclination diverging from a point in the plane of 

 reference, and lying in one and the same perpendicular plane, 

 shall converge to a focus in the plane of reference. 



Take rectangular axes of x and y in the common perpendicular 

 plane, the axis of x being in and the axis of y perpendicular to 

 the plane of reference. Let s denote distance measured along a 



dx 

 ray ; then — or cos 6 is sensibly equal to unity. Also to the 



same degree of approximation we have 



0= 



: tan 6= 



z d JL 



dx 



} 



1 _ 



p~ 



dO 

 ds 



= — 



d6 

 dx 



dx r 



(0) 



The problem which we have to solve has therefore been reduced 



to the following :— find what function -t-| must be of y that the 



increment of x from y = to the next occurrence of y = shall 

 be independent of the maximum value of y. Mathematically 

 considered, this is precisely the problem of finding what law of 

 acceleration for a particle executing vibrations about a position 

 of equilibrium will render the vibrations isochronous, y denoting 

 the distance of the particle at time x from the position of equi- 

 librium. Its solution is well known to be 



^ = -1 (D) 



dx* a* K ] 



a being a constant; and the required law of index is therefore 



i- (K) 



dlog/j, __ 



dy a 1 



The general equation of a ray in the plane of x~, y will be the 



M2 



