42 REPORT— 1872. 



iure approaches to a maximum or to a minimum as the direction of the ray approaches 

 t oivards horizontality. 



Conceive two laminae, Lamina 1 and Lamina 2, each of the thickness X. Con- 

 ceive the density in each as being- constant, but that there is a sudden increase of 

 density in passing from the one to the other. Then 

 the ray of light P A will at A be suddenly bent or Fig. 1. 



deflected from its previous line. This case may be sub- 

 stituted mathematically, when the laminas are taken 

 infinitely thin, for what actually occurs in the atmo- 

 sphere. 



Now in the atmosphere the deflection of the ray of 

 light in passing from the middle of one lamina to the 

 middle of the next, as from D to E, is evidently propro- " 



tional to the thickness assumed for the lamina3, the thickness being small. Hence, 

 if we take 6 to represent the angle of deflection at A, we must bear in mind that S x X 

 for any given angle of incidence, or that S must be infinitely small when the lamina 

 IS infinitely thin. Let the angle of incidence P A B= i. Then, by the ordinary law 

 of refraction assumed as applicable to this case, 



sin i=}x sin («-5), 



in which jx denotes the index of refraction for passage of a ray from one lamina to 

 the next when the thickness of the laminte is X. 



Hence — -_=sin i cos 8 — cos i sin b, or by dividing by cos i, 



tan i , • IV . A 

 • — ■ =tan I cos S— 'Sm S. 



But S must be Infinitely small, the laminaj being infinitely thin. Hence for infinitely 

 thin lamiufe we have sin 8=S, and cos 8= 1. Hence the previous equation becomes 



tant , . . 

 ■ = tan t~6, 



1 



or 8= '^ — - tan i. 



Let D E, or its equal P A, the laminre being infinitely thin, be denoted by s. Then 

 s=:X sec('. 



Let the radius of curvature of the ray of light, or the radius of the circle touching 

 the ray in the points D and E, be denoted by R, and then we have 



Curvature '= =rr=^ — 

 It s 



Hence Curvature=^ ^", 



II X sect 



or Curvature =-^i^ sin/. 



IxX 



But since the curvature of the ray of light is independent of the small thickness 

 which we may take for the infinitely thin lamina, and can only vary with the angle 



of incidence i, we must have ^^—^ in the foregoing equation constant ; and so we 



have 



Curvature a sin i, 



which has its maximum value when i is a right angle ; that is, when the ray is 

 passing horizontally, or infinitely nearly so. 



This shows that if the ordinarily assumed law of refraction be ti-uly applicable to 

 a ray of light passing extremely nearly horizontally through level laminaj of air of 

 varying deusitj^, the curvature of the ray of light must approach to a maximum as 

 the inclination of the ray approaches to horizontality. From this, if true, the step 

 is natural, or inevitable, to the conclusion that, leaving out of account the rotundity 

 of the earth, and conceiving the laminte of constant density to be level planes, a ray 



