252 Prof. Everett on the Optics of Mirage. 



As )jl cannot be less than unity, the law can only hold as far as 

 r = C. Within this distance, a ray, starting at any point, in a 

 direction normal to the radius vector drawn to it from the centre 

 will describe a circle about the centre. A ray starting at any 

 other angle will describe an equiangular spiral having the centre 

 for pole. 



In any medium in which the surfaces-of-equal-index are con- 

 centric spheres, if we denote by 6 the angle at which a ray cuts 

 the surface-of-equal-index at the point considered, andbyjo the 

 perpendicular from the centre on the tangent to the ray at this 

 point, we have 



I 1 dp a p 



- = - -f, cos 0= -• 

 p r dr r 



Hence, by the law of ray-curvature, we have 



1 dp_ dfogji „*__! dfip 



— — — — — COS U — - * j y 



r dr dr fi dr r 



or 



dp _ dp, 

 p ~ p' 

 or 



fip=Q 3 



a well-known result, usually obtained by regarding continuous 

 variation of index as the limit of an indefinite number of small 

 changes per saltum. 



XIV. When the surfaces-of-equal-index are coaxial circular 

 cylinders, the radius of any cylinder being called r, the value of 



— ~— is the product of — by the secant of the angle at which 



a ray cuts the surface-of-equal-index at the point considered ; 

 and this product will be the same for all rays at the same value 

 of r. If the ray be a helix described on one of the cylinders, 



the angle in question is zero, and the curvature — of this helix 



d \o°" LL 



will be equal to the value of — -j^— for this cylinder. 



Let the equations of the helix be 



x x 



y = r cos— , 2 = rsin— • 

 * a a 



1 . r 



Then the value of — is found to be — } -<,. Hence we have 



p a z + r z 



d log //, _ r 



~dr~~~ ~<^+7 2; 



