the received Theory of Newton's Rings. 455 



where a represents the quantity of light reflected at the upper 

 or lower boundary of the layer, d the depth of the layer, and 

 r the angle of refraction from glass to air. For the trans- 

 mitted light, on the contrary, where no reversion of the vibra- 

 tion takes place, the intensity is 



t/ . ■ a „ d cos r ,. N 



J'=l-4asin 2 27T — - — , .... (4.) 



A, 



so that the intensities are complementary, the sum of both 

 being the intensity 1 of the incidental light. 



For reflected light, we obtain from (3.) the maxima of in- 

 tensity for 



X 3X 5X 7X 



4 cos r 9 ~~ 4 cos r 9 4 cos r 9 ~~ 4 cos r 9 



and the minima of intensity for 



2X AX 6X 



d=0, ==■ 



3 r 9 4 cos r 9 4 cos r 9 



because for the first series J = 4<z, and for the second J = 0. 

 For the transmitted light, we obtain from (4.) 



the maxima for 



, n 2X 4A- 6X 



tf = 0, =- , = - , ==- , 



4 cos r 4 cos r 4 cos r 



and the minima for 



X SX 5X 7X 



d= 



4 cos r 9 4 cos r 9 



because for the first series J'=l, and for the second series 

 J'=l— 4a. Now as the diameters or radii of the coloured 

 rings are proportional to the square roots of the depths of the 

 layers of air, the diameters or radii of the maxima in the case 

 of reflected light bear to each other the ratio Vl : V3 : \/5...\ 



and those of the minima the ratio ^0 : */2 : a/4 With 



transmitted light, on the contrary, the ratios of the maxima 



will be \/0: */2: \/4 , and those of the minima Vl : 



VS: s/5 



This is the manner in which Newton first expressed the 

 law, and in the same manner it has heretofore been deduced 

 from the undulation theory. It is however so far from agree- 

 ing with the same, that the law, to be rendered applicable to 

 the centre of the ring-system, would require entire reversion. 



The rings nearest the centre begin to lose their circular 

 shape, and to assume one more or less elliptical, immediately 

 before the appearance of the constant dark spot, and while 

 the centre is still bright. There can be no doubt that actual 



