Form of Newton's Rings. 235 



Also B = -[{x*+f) sin 0-2**], 



C = -2y cos 6 (z + z x ). 



We have hitherto not considered the effect of the term in- 

 volving cf>, but since the minimum of B 2 + C 2 is not always 

 zero, we cannot pass these terms over without notice. 



When y = 0, and x sin 0—2^ = 0, B and C vanish, and there- 

 fore the points on the line in the plane xz are always formed 

 clearly. 



The only reason why we cannot see rings of a very high 

 order in this plane is that the rings of different colours over- 

 lap to such an extent as to obscure the result if the incident 

 light be not strictly monochromatic. 



When x = 0, and z=—z l , C vanishes, but B does not; in 

 fact, 



B = */sin0, 



k 

 = - hr\ tan 6 ; 

 r 



since ?/ 2 = a 2 = ^, when * = 0. 



J cos 0' 



Therefore, the distinctness depends on the magnitude of 

 hX tan 6. This quantity increases as h increases, and there- 

 fore the outer rings in the direction of the axis of y are always 

 less distinct than the inner. Further, as increases, tan 

 increases ; and when the incidence is nearly grazing, very 

 few rings can be formed distinctly, since small variations of X 

 are sufficient to cause large variations of B, and taking also 

 into account the variations of <f>, the colours overlap so as to 

 cause all trace of variation of intensity or colour to disappear. 



Since B 2 + C 2 is unchanged by the substitution of — y for y y 

 the rings are equally distinct on the two sides of the plane 

 xz. 



Again, since 6 lies between and — , when //, is greater than 



z 



unity (*. e. in general) z x is positive. Hence 

 (* 2 + ?/ 2 ) sin 6 + 2xz } 



is decreased by the substitution of — x for x. 



Thus the part of the rings on the side of the plane yz 

 which is nearest the incident light is less distinct than the 

 other part. 



With the help of the equation of the " surface of inter- 

 ference," we find that 



T 2 



