Prof. Everett on the Optics of Mirage. 165 



a point in the plane of maximum index; but the law of index 

 there deduced involves similar consequences for rays emanating 

 from any other point ; for the substitution of x + ira for x in 

 equation (F) merely changes the sign of y. Hence rays diver- 

 ging from a point (x, y) will converge to a focus at the point 

 [x + ira, —y) } then to another focus at the point [x + Znra, y) } 

 and so on. 



It thus appears that in a medium in which the law (E) pre- 

 vails, every object will yield a series of real images, alternately 

 inverted and erect, ira being their common distance asunder in 

 a direction parallel to the plane of maximum index; while, as 

 regards distances measured normal to the plane of maximum 

 index, each image in the series corresponds to the reflected 

 image of its predecessor with respect to this plane. It is of 

 course to be understood that the images are formed in one di- 

 mension only, like those formed by a cylindrical lens. 



I may remark incidentally that in a medium in which the 

 surfaces-of-equal-index are parallel planes, if one ray of small 

 inclination to these planes is a curve of sines, all rays of the 

 same or less amplitude must also be curves of sines; for equa- 

 tion (D) cannot hold for one ray unless (E) holds for all dis- 

 tances from the plane of reference not exceeding the amplitude 

 of that ray. 



I may also remark that a prism-like or lens-like arrangement 

 of surfaces-of-equal-index produces less deviation in rays than 

 an arrangement in which these surfaces are approximately pa- 

 rallel to the course of the rays. This is obvious from equation 

 (B), which shows that, for a constant rate of variation of logyu, 

 normally to the surfaces-of-equal-index, the curvature is propor- 

 tional to cos 6. 



The fact that rays emanating from any point in the medium 

 converge to a focus, and that the focal length measured parallel 

 to the axis of x has a constant value, corresponds to the self- 

 evident proposition in cycloidal oscillation, that the time from 

 any point to the symmetrically situated point on the other side, 

 when one of the extreme positions is taken in the interval, is the 

 half-period of oscillation. 



If, instead of supposing the surfaces-of-equal-index to be 



parallel planes, we suppose p, to be a function both of y and x, 



d 9 y 

 then -~ will be a function both of y and x, and the analogous 



supposition in cycloidal oscillation is that of gravity varying 

 with time. On this supposition, it is clear that if two particles 

 start at the same instant with different velocities from the lowest 

 points of two equal cycloids, they will keep time with each other, 

 however great and sudden the variations of gravity may be sup- 



