Modification of Newton's Rings. 283 



This stream of light interferes with that reflected from the surface 

 of the metallic reflector: in which the expression for the displace- 

 ment is a.A.^(vt-x'-B). In this formula *' differs from a- 

 by the difference of paths described by the two rays, or rather by 

 the space in air equivalent to that difference; which, if T be the 

 thickness of the plate of air, is 2 IT cos ,'. Thus the displacement 

 by the metallic reflexion is aA.sm ^ (vt-x- % Teas i'-B), wh ere 

 A and B are probably functions of .' whose form is unknown 

 We can assert however that when .' is small, A is positive and 

 B ,s not great (else, as will be seen from the subsequent ex- 

 pressions, there would be a white spot at the center) : and the 

 gradual change of appearances, except at the polarizing angle, 

 makes it probable that B is always small. At any rate there 

 is not the slightest reason to believe that A or B, which depend 

 only on the properties of the metal, would undergo any sudden 

 change exactly at the polarizing angle of the glass. 



Now the peculiar phamoinenon which is the principal subject 

 of this paper is thus explained. The central spot, where T=o, 

 is produced by the composition of two displacements 



tan (<-<') 2tt , 



a toT(7T7) sln T ( *"-*) 



and a. A. sin— (vt-x) (considering B = o). 



As ,' is greater than ,, the former of these is negative when , 

 is small, and the latter is positive. Consequently these displace- 

 ments partly or entirely destroy each other, and the center is 

 dark. But at the polarizing angle, « + .' (by Brewster's law) =90% 

 and tan (, + ,') is infinite; the first expression vanishes, and the 



NN 2 



