164 Prof. Everett on the Optics of Mirage. 



general integral of (D), namely 



y =b *' m ^r> ( F ) 



representing a curve of sines cutting the axis of x at points 

 whose distances from the origin are 



c, c + ira, c-\-2ira,... 



Th& values of b and c vary from one ray to another ; but a is 

 the same for all; and hence the distance ira between two conse- 

 cutive intersections of a ray with the axis of x is independent of 

 the amplitude b. This constant quantity iva is evidently the 

 focal length; and rays of small inclination diverging in the plane 

 x, y from a point in the axis of x will converge to a series of foci 

 at this constant distance apart. The same reasoning which 

 proves that all small vibrations are isochronous, proves that 

 wherever a plane of maximum index occurs, the other surfaces 

 of equal index in its neighbourhood being also parallel planes^ 

 rays. of small inclination diverging from a point in this plane 

 must have a constant focal length ; and it can be shown that the 

 smallness of the aberration of rays from this geometrical focal 

 length is especially promoted by symmetry of the medium about 

 the plane of maximum index. 



In fact, if we suppose log//, and its differential coefficients 

 with respect to y to be continuous, the assumption that the sur- 

 faces of equal index are parallel planes gives 



log^=A + B?/'+G2/ 2 + D?/ 3 + ..., 

 whence 



^ = B + 2Cy + 3D^+... 



The assumption that y is measured from a plane of maximum 

 index gives 



B=0, 2C negative = ^ suppose. 



Hence, when y 2 is negligible^ we have approximately 

 . . d\ogp_ y 



which is identical with equation (E). The further supposition 

 that the medium is symmetrical about the plane from which y 

 is measured makes D = 0, because no odd powers of y can enter 

 the expression for log ^. 



III. The investigation in (II.) related to rays emanating from 



