OF NEWTON'S RINGS BEYOND THE CRITICAL ANGLE. 651 



niitted, than in the reflected light, in consequence of the quantity of white light reflected about the 

 edge of the spot. The separation of colours is however but slight, compared with what takes place 

 in dispersion or diff"raction, for two reasons. First, the point of minimum intensity is the same for 

 all the colours, and the only reason why there is any tint produced is, that the intensity approaches 

 more rapidly to its limiting value 1 in the case of the blue than in the case of the red. Secondly, 

 the same fraction of the incident light is reflected at points for which Z) »= X, and therefore r cc ,y/\ ; 

 and therefore, on this account also, the separation of colours is less than in diffraction, where the 

 colours are arranged according to the values of X, or in dispersion, where they are arranged according 

 to values of X"^ nearly. These conclusions agree with observation. A faint blueish tint may be 

 perceived about the dark spot seen by reflection ; and the fainter portions of the bright spot seen 

 by transmission are of a decided reddish brown. 



16. Let us now consider the dependance of the size of the spot on the nature of the polarization. 

 Let s be the ratio of the intensity of the transmitted light to that of the reflected ; «„ s,, the par- 

 ticular values of s belonging to light polarized in the plane of incidence and to light polarized 

 perpendicularly to the plane of incidence respectively ; then 



, / sin 2 \ ' , 



I 



Now according as s is greater or less, the spot is more or less conspicuous ; that is, conspicuous 

 in regard to extent, and intensity at some distance from the point of contact ; for in the immediate 

 neighbourhood of that point the light is in all cases wholly transmitted. Very near the critical angle 

 we have from (10) S2= n^s,, and therefore the spot is much more conspicuous for light polarized 

 perpendicularly to the plane of incidence than for light polarized in that plane. As i increases the 

 spots seen in the two cases become more and more nearly equal in magnitude : they become exactly 

 alike when i = i, where 



sin% = . 



When i becomes greater than i the order of magnitude is reversed ; and the spots become more 

 and more unequal as i increases. When 1 = 90° we have «, = fi*S2! so that the inequality becomes very 

 great. This however must be understood with reference to relative, not absolute magnitude; for 

 when the angle of incidence becomes very great both spots become very small. 



I have verified these conclusions by viewing the spot through a rhomb of Iceland spar, with its 

 principal plane either parallel or perpendicular to the plane of incidence, as well as by using a 

 doubly refracting prism ; but I have not attempted to determine experimentally the angle of inci- 

 dence at which the spots are exactly equal. Indeed, it could not be determined in this way with 

 any precision, because the difference between the spots is insensible through a considerable range of 

 incidence. 



17. It is worthy of remark that the angle of incidence i at which the spots are equal, is exactly 

 that at which the difference of acceleration of phase of the oppositely polarized pencils, which arises 

 from total internal reflection, is a maximum. 



When i = I wc have 



2/u 

 sin 20 = sin 'Jd) = — — -; whence cot() = tan rf) = /i...(l l) ; 

 '^ /i' + I '^ 



andp^^ K*,L,r,. ^ where ry = e ^^"'^■...(12). 



