162 Prof. Everett on the Optics of Mirage. 



Hence it is easily shown by similar triangles that a very short 

 line dN drawn from either of the points towards the centre of 

 curvature, is to the whole length p of which it forms part, as dv 

 the difference of the velocities of light at its two ends is to v the 

 velocity at either end. That is, 



p v 



the negative sign being used because the velocity diminishes in 

 approaching the centre of curvature. But, since v varies in- 

 versely as fju, we have 



dv __ dfi 



v fl 



Hence the curvature - is given by any one of the four following 



expressions : — 







1 _ I dv _ 



tZiogv 1 dfju 



dlogp, 



p ~ v dN 7 



dN ~ p dN 



dN 



• (A) 



This proof is due to Professor James Thomson, who gave it 

 in a paper to Section A. at the recent Meeting of the British 

 Association at Brighton. 



It is obvious that the osculating plane is the plane which con- 

 tains the direction of most rapid increase of index at the point 

 considered, and that it cuts at right angles the surface-of-equal- 

 in'dex drawn through the point. 



Cor. 1 . The curvatures of different rays at the same point are 

 directly as the rates of increase of /n in travelling along their 

 respective normals. If 6 denote the angle which any ray makes 

 with the surface-of-equal-index at the point, or which the radius 

 of curvature of a ray makes with the direction of most rapid in- 

 crease of index, the curvatures will be directly as the values of 



cos 6. In fact, if -— denote the rate at which a increases in a 

 ar 



direction normal to the surfaees-of-equal-index, we have 

 dfi dp a 



and therefore 



1 1 dfi ' dlog/j, a 



- = - -y- cos 6 — — ~- cos 6. . . . B) 



p p, ar dr v ' 



Cor. 2. When & is small, and is regarded as a small quantity 

 of the first order, cos 6, being approximately l~\6 2 , differs from 

 unity by a small quantity of the second order. Hence cos 6 is 

 sensibly constant when 6 is small, and rays which are but slightly 



