650 Mb. stokes, ON THE FORMATION OF THE CENTRAL SPOT 



13. For the transmitted light we have an expression similar to (G), with — nx: in place of nz, 

 and a different coefficient C^, where 



l-eV 1-96 b^-qb 



When the light is polarized in the plane of incidence we have 



- y/- 1 . Sq sin 20 

 ' (1 - g-)cos20 - -v/~(l +9') sin2 



_ 29 sin 20 1(1 + q^) singg- y/- 1 (1 - q') cos 29\ 

 (1 - q-y + iq'sm-Qd 



so that if \|/^ and p^ refer to the transmitted light we have 



•(7); 



2_ 4(/^sin^20 

 '''" (1 - q'y + 4:fsm'2e 



If we take p, positive, as it will be supposed to be, we must take \p-^ such that cos \|/^ may be 

 positive ; and therefore, of the two angles lying between - ir and tt which satisfy (8), we must 



choose that which lies between and +— . Hence, since from (3) and (S) \|/^ is of the form 



\j, + - + JiTT, n being an mteger, we must take x|/^ = \^ . 



For light polarized perpendicularly to the plane of incidence we have only to put (p for d. It 

 follows from (4) and (g) that the sum of the intensities of the reflected and transmitted light is equal 

 to unity, as of course ought to be the case. This renders it unnecessary to discuss the expression 

 for the intensity of the transmitted light. 



14. Taking the expression (4) for the intensity of the reflected light, consider first how it varies 

 on receding from the point of contact. 



As the point of contact D = 0, and therefore from (5) q = 1, and therefore p' = 0, or there is 

 absolute darkness. On receding from the point of contact q decreases, but slowly at first, inasmuch 

 as D varies as r-, r being the distance from the point of contact. It follows from (4) that the 

 intensity p' varies ultimately as r% so that it increases at first with extreme slowness. Consequently 

 the darkness is, as far as sense can decide, perfect for some distance round the point of contact. 

 Further on q decreases more rapidly, and soon becomes insensible. Consequently the intensity de- 

 creases, at first rapidly, and then slowly again as it approaches its limiting value 1, to which it soon 

 becomes sensibly equal. All this agrees with observation. 



15. Consider next the variation of intensity as depending on the colour. The change in 9 and 

 cf) in passing from one colour to another is but small, and need not here be taken into account : the 

 quantity whose variation it is important to consider is q. Now it follows from (5) that q changes 

 the more rapidly in receding from the point of contact the smaller be X. Consequently the spot 

 must be smaller for blue light than for red ; and therefore towards the edge of the spot seen by 

 reflection, that is beyond the edge of the central portion of it, which is black, there is a predominance 

 of the colours at the blue end of the spectrum ; and towards the edge of the bright spot seen by 

 transmission the colours at the red end predominate. Tlie tint is more conspicuous in the trans- 



