of Light hj thin Layers of Metal. 459 



|5= sin* (e+uHU^D-' + V^D*) 

 + 2 cos 2 (e + u) (A cos 2L + B cos 2L) . 



For A = 0, ^— r has the positive initial value 



But the second differential quotient can never be negative for 

 e + u > -r , because we have 



-2VU 1 U 3 V 1 V3)>0. 



The numerical value of the parenthesis Acos2L + B cos2L 

 therefore cannot exceed the limiting values , 



± */A 2 + B 2 = ±2^(U 1 U 3 V 1 V 3 ). 



Therefore constantly 



(U 1 U 3 D- 2 + V 1 V 3 D 2 )±2(Acos2LH-Bcos2L)>0, 



whence ^- r? >0 results for e + u> — . But hence it further fol- 



lows that §-r-, starting from its positive initial value, constantly 



increases, and so can never become 0, and therefore that, in the 

 light that passes through, there are no maxima or minima, but 

 the intensity continuously diminishes as the thickness increases. 

 For the reflected ray, on the contrary, the condition for the 

 maxima and minima assumes the form 



U 3 D- 2 sin(2L + e + ^-^)) + V 3 D 2 sin(2L-e-w-^) 

 sin2(e+w)=0, 



-2A / U » V » 



where 



ta ±+*-_^L n> tant^ = 4zi cot(€+M) . 

 & 2q 3 sm (e + u) * q\ + 1 



For greater values of A the influence of the term multiplied 

 by D -2 becomes here evidently preponderant; and since this, in 

 consequence of its periodic factor, infinitely often changes its 

 sign, an infinite number of maxima and minima exist. These, 

 however, diminish in distinctness very quickly, since the periodi- 



