312 Mr. R. B. Sangster on some Consequences of 



•+-% value furnishes real values of tan i in the range of i 

 under consideration. 



In the case of /u,<l the conditions for employing the + 

 or ~~ X va ^ ue are reversed. In showing this, it has first to 

 be noted that 1/n can only have a meaning when it is taken 

 as positive, and that fi being < 1, (/*+ 1 )/(/* — 1) is negative. 

 Therefore, let >/n= (1 +/*)/(l— /a), and we find the +% value 

 = /^ _1 and the — % value =/jl. Substituting these values of 

 X as before, sini = respectively and fi(jJL 4 — 1)^/(/jL q — 1)2, 

 the two values of % thus occurring in an order opposite 

 to that which held in the case of /«t>l. Also, in the remain- 

 ing range of i, from where 8' — /*S to total reflexion, it 

 is easily shown that the — % value has to be adopted. 

 Similarly, it can be shown that the adoption of the — % value 

 furnishes real tangent values for this range of i when /jl< 1. 



Therefore, when the light vibrates in the plane of in- 

 cidence, in order to reflect l//*th at a plane interface, the 

 angle of incidence is determined by (3) or (1) where 

 % =(\/n + l) 2 /(n — 1). When yLt>l, the minus sign applies 

 to the range of i from normal incidence to tan -1 //, and the 

 plus sign to the range of i from the latter angle to grazing 

 incidence. This may be referred to the value of n by stating 

 that when n< (/x + l) 2 /(/i,— l) 2 the +% value should be 

 adopted, and when 



n=or>(,* + l)70-l) 2 , 



then both % values may be employed, thus determining two 

 incidences where the reflected light is equal. When //<<1, 

 the order of adoption of these signs has to be exactly 

 reversed. 



The reflexion at one interface only has, so far, been dealt 

 with, but towards the end of the paper it will be shown how 

 to determine i in order to reflect or transmit a given ratio of 

 the incident light when both surfaces of a plate are involved. 

 To provide a reference for that case, it is here necessary to 

 show that when 1/nth the incident light is reflected at the 

 first surface of a homogeneous isotropic refracting medium 

 bounded by parallel planes, then 1/nth of the refracted light 

 is reflected at the second surface. 



If /u be the refractive index of the second medium with 

 respect to the first, /a -1 is the index of the first with respect 

 to the second. For convenience, let a single value of ^ in 

 terms of n be employed for the incident ray, but we must 

 remember the alternative ^ value must then be adopted for 

 the refracted ray. We have, then, 1/nth light incident on 



