HOMOGENEOUS FLUIDS BY POLARIZED LIGHT. 59 



parallelly and perpendicularly to the principal section S S'. It 

 is sufficient to consider those which concur in the formation of 

 one of the images, the extraordinary image for example. Their 

 intensities are represented by the following expressions : 



P.O.N.R.S. +lsinr.cos{i-r).F„+^, 



P.O.N.T.S. +^cosr.sin(i-r).F„+i, 



P.O.P.R.S. +icosr.cos {i-r).F„+i, 



P.O.P.T'.S'. -^sinr.sin(i-r).F„, 



P.E'.N'.R'.S'. -^sinr.cos(i-r).F,+i, 



P.E'.N'.T'.S'. -~cosr.sm(i-r).Fe+i, 



P.E'.P.R.S. +-Jcosr.cos {i-r).Fe+i, 



P.E'.P.T'.S'. -^sinr.sin(i-r).Fe. 



Adding the expressions which have the same characteristic, 

 and observing that - in the characteristic is equivalent to the 

 minus sign, the eight pencils are reduced to four : 



— — sin r [cos (^ — r) + sin {i — r)'] Fg, 



+ -cos )• [cos {i—r) + sin {i~r)] F„+i, 

 + — sin r [cos {i — r) — sin (i — r)'] F^, 



+ — cosr [cos {i—r) — sin (i — r)] Fe+^. 



On inspecting these formulae, it is seen at once that the image 

 passes to white when i — r = 45°, for then the two last pencils 

 disappear ; the intensity of the light becomes independent of the 

 difference between e and o, and consequently is the same for 

 every kind of rays. The colour of the image attains, on the 

 contrary, its highest degree of vividness when i — r is equal to 

 zero or to 90°, that is to say, when the principal section of the 



