458 M. E. Jochmann on the Reflection and Refraction 

 hand, the result is remarkable in a theoretical point of view, that 



IT 



the system of formulas corresponding to the case e=— is almost 



perfectly identical in form with that above exhibited for total re- 

 flection by transparent lamellae — namely, with the modification 

 that the numerators of the expressions for tan £>' and tan b m 

 receive the opposite signs. The condition s 2 >l, which limits 

 the variation of the angle of incidence, disappears in this case, 



7T 



and s l may increase from to — , unless the greater values be 



excluded by the occurrence of total reflection at the third medium. 

 When A = oo, we have the case of a total reflection for any angle 

 of incidence. It is easy to prove that, with increasing thickness, 

 neither in the refracted nor in the reflected light can there be 

 maxima or minima of intensity ; for the maxima of a n and a' 2 

 would necessarily corespond to the minima of a n f and a y/ / 2 , and 



also to the maxima of the expression ( D — ~r ) + 4 sin 2 ((f> l + $3)* 



But the second term in this is independent of the thickness ; 

 while the squared parenthesis possesses, between the limits A = 

 and A=oo , no real minimum. Such a medium, therefore, in a 

 layer of variable thickness, would not exhibit, either in the re- 

 flected or in the transmitted light, an interference-phenomenon 

 analogous to Newton's colours of transparent bodies. 



For the transmitted ray this assertion may be at once gene- 

 ralized thus far : — with increasing thickness maxima and minima 



TT 



of intensity can never occur, provided that e + u >• — . For let 



us imagine both numerator and denominator of the general ex- 

 pression for a ul 2 divided by D 2 ; the numerator becomes inde- 

 pendent of the thickness, and the denominator takes the form 



N = U J U 3 B- 2 + Y 1 V 3 D 2 -2(Acos2L + Bsin2L), 

 where, as an easy calculation shows, 



The maxima, then, of « /y/ 2 must correspond to the minima of 

 N, and vice versa. Differentiation according to A gives : — 



A_ |*L = s i n (e*«) (ILUjr 2 - V^D 2 ) 

 4ttc 2 9A . ; - is; 



-f 2 cos (e + u) (A sin 2L — B cos 2L), 



