166 Prof. Everett on the Optics of Mirage. 



posed to be*, unless these variations be such as to compel one 

 of the particles to travel beyond the cusp of its cycloid and thus 

 introduce discontinuity. 



As velocity in the case of the particle corresponds to ■—■ in the 



Q/X 



case of a ray, the corresponding inference is that, in the medium 

 now supposed, rays which proceed from the same point in the 

 axis of x at different inclinations to this axis will meet it again 

 at the same distance. The conclusion cannot be extended to 

 points above or below the axis of x. 



IV. When the surfaces- of-equal- index are parallel planes, the 

 deviation of a ray in passing from one of these surfaces to an- 

 other can be expressed in terms of the angle of incidence at the 

 first surface and the relative index from the first surface to the 

 last — being entirely independent of the distance between the two 

 surfaces, and of the character of the intervening layers. 



For, since the axis of y is perpendicular to the planes-of-equal- 

 index, equation (B) becomes 



1 _ tfTlog fM 



P ~ dy 



Hence 



0. 



d6= — — = — -cos Odlop; u= -. — 7,-^logu, . (G) 

 p dv ° r sin 6 ° " v ' 



P dy 

 or 



sin Odd 



cos 



dlogJJL= 3- =— ^ log COS ^. 



Integrating from fi i3 6 X at the first surface to /-i 2 , # 2 at the last, 

 we have 



b = ^h (H ) 



/4j cos 6^ 2 v y 



When the change of index is abrupt, this equation amounts to 

 a statement of the " law of sines ;" for cos 6 l is the sine of the 

 angle of incidence, cos # 2 is the sine of the angle of refraction, 



and — is the relative index from the first medium into the 



second. Instead of integrating between limits, we might have 

 deduced the general integral 



fju cos 6= constant, 



which applies to the whole course of the ray. 



* For such changes will not disturb the equalities, Ratio of accelerations 

 = Ratio of velocities = Ratio of distances from vertex = Constant. 



