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



This value of sin r is what we should have had for sin i, if, 

 in the first instance, we had written jul' 1 for /x, and — \/q 

 instead of + \/q as is required for /jl< 1, and we see it is the 

 sine angle of the refracted raj. Therefore, in the case of 

 common light also, when 1/wth the light incident on a plane- 

 parallel plate is reflected at the first surface, 1/nth the 

 refracted ray is reflected at the second surface. 



It will be observed that this is equivalent to saying that 

 s'm 2 (i — r) tan 2 (i— r) _ sin 2 (r— i) tan 2 (r—i) 

 sin 2 (i + r) tan 2 [i •+- r) sin 2 (r+i) ' tan 2 (r + i)' 

 an obvious enough equation; but the individual demonstra- 

 tions by means of the 1/nth formulae are also necessary in 

 view of the varying alterations of sign required in these 

 when jn is > or < 1. 



The refracted ray is reflected many times to and fro in the 

 interior of the refracting medium from surface to surface, 

 but if the reflecting surfaces are plane-parallel, the angle of 

 incidence is the same at each reflexion, and if 1/nth is 

 reflected at the first interior reflexion, 1/nth. will be reflected 

 at each successive contact with a surface. In consequence, 

 there is a simple way of determining the angle of incidence 

 in order to reflect or transmit a definite proportion of the 

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



Suppose it is required to reflect 1/mth the light incident 

 on a transparent, homogeneous, isotropic refracting medium, 

 bounded by parallel planes, where the repeated reflexions and 

 transmissions at both surfaces have to be taken into account. 

 When, under these conditions, 1/mth the incident ray is 

 reflected back into the first medium, let 1/n be the ratio 

 reflected at the first incidence on the first surface of separa- 

 tion ; what we need is the relation of ra to n. 



Fis-. 4. 





\Ag 



Referring to fig. 4, where AB and CD are the surfaces of 

 the refracting medium, (1) is the incident ray. At the first 



