64 Mr. Mac Cullagh on the Laws of 



may become impossible for certain values of n. For it is clear that if n lie 



between the limits l and — , the numerator and denominator of the fraction (69) 



will have unlike signs, and the tangent of w, will be the square root of a negative 

 quantity. In this case, therefore, if common light be incident, it will " refuse 

 to be polarised," as Brewster expresses \i ; in other words, it will ,be impossible 

 to find an angle of incidence at which the reflected pencil will cease to contain 

 light polarised perpendicularly to the, plane of incidence,,or at which the reflected 

 transversal t', will vanish. With all values of n, except those which are included 



between the narrow limits l and — , the polarising angle is possible. It is zero 



at the latter limit, and 90" at the former. Outside these limits it changes 

 rapidly at first, until n has passed either ojf them by a quantity considerable in 

 proportion to the interval between them. 



From (68) we find ■By{=-\, when a=-\, or n:=a ; and also zT{=.Q(f — A, when 

 /» = ], or n=:b. In the latter case it is remarkable that no light is reflected 

 when common light is incident at th* angle 90°— X. For then we have r'g^O ; 

 and, because ti^t^' ^^ ^&'y& likewise t3=0., /Therefore no light can enter the 

 reflected pencil. But this case deserves that we should consider it more at large, 

 without restricting ourselves to the supposition that the axis of the crystal lies in 

 the plane of incidence. 



Assuming then that n=:b, or that the refractive index of the fluid, which 

 covers the reflecting surface, is equal to the ordinary index of the crystal itself, 

 we may observe that, in this case, every angle of'fncidence, in every azimuth, has 

 ^ right to be regarded as a polarising angle. In fact, common light cannot 

 suffer reflexion at the separating surface of the crystal and the fluid, without 

 becoming completely polarised: For if polarised light be incident, and if t, and 

 t'j be, the uniradial reflected transversals, respectively belonging to the ordinary 

 , and to the extraordinary ray, the former transversal must necessarily vanish, for 



'"l.~' / the same reason that no reflexion can take place at the separating surface of two 

 ''t' . ' ordinary media whose refractive indices are equal ; and thus the actual reflected 

 ■^ transversal will always coincide in direction with t'j, whatever be the direction of 



the incident transversal. Consequently, if common light be incident, the whole 

 „ reflected pencil will be polarised in a plane passing though t'.,, and making witli 



^^. ^^\ vy.„./,. /- /_. /^^ 



/,,..,/-., /;yV- C^'-t 



/,^ '. ■a-' ' — •' '•yrr^ 



