Sir David Brewster on the Compensations of Polarized Light. 387 



Having thus determined that light polarized in a plane whose inclination to 

 the plane of reflexion is + (p, will be compensated by oppositely polarized light, 

 whose inclination is — (p, if both the lights are reflected, or by refracted light 

 whose inclination is 90° — or 0', we must next endeavour to discover at what 

 angle of incidence the polarized light submitted to the polarimeter, has suffered 

 reflexion or refraction, when we have the angle of incidence and the inclination 

 of the plane of polarization, by which we have effected the compensation. 



Let us first take the case when light partially polarized by reflexion is com- 

 pensated by the polarization produced by refraction through one surface, at an 

 incidence i of 80°. The index of refraction being 1.525, we shall have when 



or = 45°, cot 0' = cos (i — i'), and 0' = 52° 33'. 



Now, the plane of the light polarized by reflexion must be inclined 90° — 0', or 

 37° 27' ; we must, therefore, find the angles of incidence above and below the 

 polarizing angle, or the two values of i corresponding to this value of 0, namely, 

 37° 27', at one or other of which the original light must have been reflected. 

 These values will be obtained from the expressions 



cos (^ + i') , . ., sin i 



tan = -: 7pr, and sin i = . 



cos (/ — I ) m 



When ^ -|- i' is less than 90°, or when the angle of incidence is less than the po- 

 larizing angle, tan is positive, and we have 



sin i = ^/ (m'^+l)a^tan 0)-^ / ^ ^/"~7~2';;r^ ~4T^^ -( 

 8 tan I "^ W* + 1 ;< -^ (1 -tan 0)U ' 



When i + i' is greater than 90°, and tan negative, the formula becomes 



smz-y _8tan0 l~^- ^ + V^^F+lJ ^ (l+tan0)^/ ' 



From these formulae, whem m = 1.525 and =: 37° 27', we obtain i:= 24° 50', 

 and 83° 30'. 



When the compensation of refracted light is effected by one reflexion, either 

 above or below the polarizing angle, for example, at 15° 40', and 87° 51', we 

 shall have 



3d2 



