250 Prof. Everett on the Optics of Mirage. 



and its direction-cosines (regarding it as drawn from the centre 

 of curvature) are 



rt d*y d 2 z 



°> - p d^' ~ p dx~ 2 ' 



Now the line r is in the osculating plane of the ray at the point 

 x, y, z, because the osculating plane always contains the direc- 

 tion of most rapid change of index ; hence the lines r and p are 

 coincident, their common direction being defined by the inter- 

 section of the osculating plane with a plane perpendicular to the 

 axis of x. Their direction-cosines are therefore equal ; that is, 



y _ d 2 y z _ d 2 z 

 r " dx 2 r p dx 2 



But we have 

 Therefore 



1 _ dlog /jL _ r 



dr a 2 



d 2 y _ y 1 _ y d 2 z _ z 

 dx 2 r p a 2 ' dx 2 ~ a 2 



But it is clear, from Section II. of the previous paper, that the 



d v v 



general differential equation of a ray in the plane xy is ^-|= — -^. 



Hence the projection of a ray upon the planer?/ is identical with 

 the path of a ray in the plane xy ; or, to state the same thing in 

 general terms, the projection of a ray upon any plane containing 

 the axis of x is identical with the path of a ray in this plane. 



The equations of the projections of a ray upon the planes xy, 

 xz are 



. . X ~~ — C 7 i . X ~~ ' c 



w=osin , z=o' sm 1 



y a a 



b, c, V, c 1 being arbitrary constants. They show that the ray is 

 in general a helicoid or flattened helix, capable of being inscribed 

 in a rectangular tube whose section has the length and breadth 

 2b and 2b f . 



As the substitution of x + ira for x merely changes the signs 

 of y and z, every object in the neighbourhood of the axis of x 

 will yield a series of real images alternately inverted and erect, 

 their common distance apart measured parallel to the axis of x 

 being ira. The pencil of rays proceeding from any point to its 

 conjugate will in general consist of helicoids, both right-handed 

 and left-handed, of every degree of flatness, from the true helix 

 to the plane curve of sines. If the point is on the axis of x, all 

 the rays will be plane ; and even if the surfaces-of-equal-index 

 be not circular cylinders, provided only they be surfaces of revo- 

 lution about the axis of x, rays emanating from a point on this 



