Prof. Everett on the Optics of Mirage. 249 



equal-index are circular cylinders having for their common axis 

 a line of maximum index, the law of index-variation in the im- 

 mediate neighbourhood of this axis must in general be 



d log fi r 



dr a 2 



r denoting distance from the axis. Hence it is clear, from 

 Sections II. and III. of the preceding paper, that rays diverging, 

 at small inclinations to the axis, from any point in or near it, 

 and lying in one plane which also contains the axis, will con- 

 verge to a series of conjugate foci, whose common distance mea- 

 sured parallel to the axis is ira. We shall now show that this 

 property is not confined to rays lying in the same plane with the 

 axis, but extends to all rays of small inclination to the axis. 



Employing rectangular axes of x, y, z, and making the axis 

 of x coincide with the line of maximum index, it follows*, from 

 the supposed smallness of the inclination of the rays to this line, 



d*x .... 1 



that j-j- is negligible in comparison with the curvature — , and 



that, to the same degree of approximation, 



<Py_d?y d*z = d*z 

 ds* ~~ dx*' ds* ~ dx 2 ' 

 The radius of curvature p is therefore given by the equation 



1 _(*lV.(dV\*. 



* Let x deDote the inclination of the ray, at point x,y,z, to the axis of 

 x; then we have 



d 2 x d dx d .doe, 



— — = — — = — cos cc = — sin a. — . 

 ds 2 ds as as ds 



But -^ is of the same order of magnitude as - : hence — % is very small 

 ds p ds 2 



in comparison with - — that is, with the square root of 



\ds*) W/ W) ' 



We may therefore write 



i=&\\( d ^y 



p 2 \dsV ^KdsV ' 

 The general expression for - ~ is 



dx 2 



d?-y 



l 2 y (dx\ 2 dy d 2 x 

 lx 2 \dsl dx ds 2 



of which the first term is, in the present case, equal to t~ 2 , and the second 

 vanishes. 



