FresneVs Reflexion of Light Theory. 321 



incidence 1/nth is reflected and 1 — ljn enters the second 

 medium ; l//ith of the latter, or, (n— l)/n 2 of (1), is then 

 reflected at the second surface to again fall on the interior of 

 the first surface, where 1/nth of (n — l)/n 2 is reflected and 

 (n—l)l-n? — (n — l)\r? is transmitted to augment the l/«th 

 reflected at first incidence. The interior reflected ray returns 

 repeatedly to the first surface where the quantity trans- 

 mitted back diminishes n 2 times each time in succession. 

 The result is that 



1 1 (n-lf Q-l)' (n-l)» 



where t is the total number of reflexions to be taken into 

 account. The sum of the series is (2n— n + l)\n(n + 1), and 

 if t is large it will not be an easy problem to determine n in 

 terms of m. But, if t is large, it is better to write £ = oo , 

 when the sum becomes 2/(n+l), whence w= 2m — 1. This 

 can be done without appreciable error when the incidence is 

 not too oblique, because all the terms to infinity after that 

 involving t only sum-up to 



2 2ft*-w+l = (n-iy 



n~+l ~ n*(n + l) n ( (n 2 -l)' 



a sum which is less than the last single term required when 

 n 2 >2. 



Therefore, when l/?nfh the incident light is required to be 

 reflected from a refracting plate, the corresponding ratio 

 required to be reflected at first incidence is l/(2m— 1), and 

 substituting the value of the denominator of this expression 

 for n in the appropriate value of %, we can determine the 

 necessary angle of incidence. 



At normal incidence, the ratio of reflected light at first 

 incidence is l/rc = (//, — l) 2 /(/x+ l) 2 . but if we add the effect of 

 the innumerable to-and-lro reflexions in the interior of a 

 plate, the total quantity reflected back into the first medium 

 is l/m = (fA— l) 2 /(yLt' 2 + 1) . At normal incidence, for fj,= 1*5, 

 l/w=l/25, while l/m=l/13. When 



tan i = /x, l/m = (> 2 - l) 2 /(^ 4 + 1). 



The incidence necessary in order to transmit a given 

 ratio through the phite can be found in a similar manner. 

 "Referring to fig. 4, we get the series 



^ _ fa-i y („-!)> (n-lf (n-1) 2 



I 2 4. I* ft I • • • • I /" ? 



where 1/m' is the ratio of transmitted to incident light and t 

 Phil. Mag. S. 6. Vol. 22. No. 128. Aug. 1911. Y 



