Prof. Everett on the Optics of Mirage. 167 



V. Putting 6 9 — in equation (H), we have 



^=COS0.. 



Ml 



Hence a ray entering at any angle 6 i to the parallel planes-of- 

 equal-index will become parallel to these planes when it has 

 penetrated as far as the plane in which 



a=yu, i cos^ 1 (K) 



It will then be bent back symmetrically, and will emerge again 

 from the plane at which it entered, making the angle of emer- 

 gence equal to the angle of incidence. This result can only 

 occur when the original course of the ray is from greater index 

 to less. 



When there are two regions of constant indices fjb l} /^ 2 (/^ 

 being the greater) separated by a region in which /uu diminishes 

 continuously from fju l to fi 2 (the surfaces-of-equal-index being 

 parallel planes), a ray entering this intermediate region from the 

 side where \l is greatest will be able to get through if cos 6 l is 



less than — . But if cos 9, is greater than — , there will be a 



plane in the intermediate region in which equation (K) will be 

 satisfied, and the ray will be returned from this plane. When 

 the change of index is abrupt, the above statement resolves itself 

 into the usual formula for the " critical angle " of total reflection. 



VI. Thus far we have been supposing the surfaces-of-equal- 

 index to be plane. If we now suppose them to be horizontal 

 surfaces parallel to the general surface of the earth, it will be 

 necessary to modify the conditions of (II.) by making the axis 

 of x not a straight but a horizontal line, which, if we regard the 

 earth as a sphere, will be a circular arc described about the 

 earth's centre; while the ordinates denoted by y will not be 

 parallel to any one line, but will be vertical, and will therefore 

 be everywhere perpendicular to the axis of x. 



Putting R for the earth's radius, equations (C) will now stand 

 thus : — 



0= tail 0=f^ 

 ax 



1 



p 



1 



*" R 



dd 



ds 



1 

 ~ II 



dO _ 



dx ~ 



1 

 R 



dhj 

 dx*' 



Ig 



for - 

 P 



its val 



dh/ 

 dx*~ 



ue — 

 1 



d\0g[M 



dy > 



d log fjb 

 dy 



we 



have 



