between two Substances of different refractive Powers. 23 



tan 9 (1/ + •") is always greater than tan 2 (/ — »") : so that 

 (1 — e*) . (1 — g 9 ) is always positive. Consequently the cen- 

 tral spot is black when e and g have different signs, and bright 

 when they have the same sign. Or as tan (< — *') is always 

 negative, and tan (»' — i") always positive, the central spot is 

 black when tan (i + *') and tan (i' + /') have the same sign, 

 and bright when they have different signs : that is, it is dark 

 when i -f- i' and r* -f- r are both less or both greater than 90°, 

 and bright when < + i' is less than 90° and / + » w 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 me- 

 dium, 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 g = 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 we have °~~ = 0, whence g = e: and the 



(2 e \ 3 

 - — -g J . The con- 

 dition g = e gives 



tan(/ — \!') __ tan (i— V) 

 tan (■' + •") " tan (< + /]' 

 whence sin 2 2 i' = sin 2 1 . sin 2 i r/ : 



or cos 9 \! = r cos i . cos i /r . 



mm 



where m and n/ are the refractive indices of the two media. 

 Without attempting to solve this equation generally, suppose 

 m as 1*53 and m = 2*45 (which correspond nearly to plate 

 glass and diamond). The values of 1'at the polarizing angles 

 are 56° 49' 54" and 67° 47' 48"; and the value of / which 

 makes the first ring black is 63° 19' 4"; the values of i and 

 \' corresponding to this are 35° 43' 57" and 21° 23' 21": 

 whence e = g = 0,083215; and the intensity of the light at 

 the central spot = a 2 x 0*02732. 



But to obtain a practical idea of the import of this expres- 

 sion we must compare it with the intensity of light in the rings 

 in some other position. Now when the incidence is perpen- 

 dicular, the expressions above give for the difference of the 

 light in the dark spot and bright rings, a a x 0*28159. Con- 

 sequently the intensity of light in the rings seen between the 

 two polarizing angles is less than one tenth of that in the rings 



