of Newton's Rings. 413 



and when T= , , &c. the intensity is 



4 cos < 4 cos i J 



VI -gel 

 and the excess of the latter above the former is 



4eg(l -<?)(!- g>) 



(i-<?g*y • 



This is the difference of intensity of the brightest and of the 

 darkest parts of the rings: and when it is positive, the center 

 of the rings is dark. 



Now tan* (< + <') is always greater than tan 2 (i-t'), and tan 2 (< ' + i") 

 is always greater than tan* (i' - ") : so that (1 — e?) . (1-g*) is always 

 positive. Consequently the central spot is black when e and g 

 have different signs, and bright when they have the same sign. 

 Or as tan(i-i') is always negative and tan(i'-i") always positive, 

 the central spot is black when tan (( + <') and tan(i'+<") have the 

 same sign, and bright when they have different signs: that is, 

 it is dark when < + <' and •' + «" are both less or both greater 

 than 90", and bright when / + «' is less than 90° and »' + <" greater 

 than 90" (or vice versa). From this it follows that while the 

 angle of incidence is less than the polarizing angle of the first 

 medium, the central spot is black : at that polarizing angle the 

 rings disappear (as e = 0): from that angle to the polarizing angle 

 of the second medium the central spot is bright: at the polarizing 

 angle of the second medium the rings disappear (as^=0); and 

 beyond that, the central spot is again dark. 



Now let us estimate the intensity of the light at the central 

 spot when the first ring is black (the angle of incidence being 

 between the two angles of polarization). If the first ring is black 



3 C2 



