Form of Newton's Rings. 229 



This expression varies least from its mean value A, as <f> 

 and yjr vary, the former from zero to a small value, the latter 

 from to 2tt, when B 2 + C 2 is least, i. e. when 



B.|? + C.|?=0, 



dz oz 



i. 6. 



2x [(> 2 + ?/ 2 ) sin 0-2xz]-±f cos 2 (z + z ] ) = 0, 

 or 



x (v*+f) sin 0-2^-2t/ 2 cos 2 (z + z l ) = 7 

 or 



2 O 2 -^ 2 ) (* + *!> cos 2 + 2a- 2 sin 2 (*-*, cot 2 0) 



- l i'sin0(> 2 + < y 2 ) = O. . (x.) 

 When (x.) holds, the value of A k in (i.) becomes 



Afc=-cos(9(u 2 + t; 2 ), 



t. e. the A of equation (ix.) 



= £ cos 0't±ll = k8, where S= — ^ (^ 2 + «/ 2 ). . . . (xi.) 



With this value, corresponding closely with that given in the 

 ordinary theory of Newton's Rings, we can determine the 

 equations of those rings at once. The ordinary methods used 

 for thin plates by Glaze brook or Airy furnish the result that 

 the equations are (for the dark rings) : — 



cos ^ t 2 i 2^ 7 ^ 



*-e. ^ 2 +/ = co ^ = « 2 (say) (xii.) 



i. e. The rings lie on this set of coaxial elliptic cylinders. 



They also lie on the surface of the 3rd order (x.). 



The principle which we have employed to determine the 

 position of the rings is both simple and satisfactory. Though 

 any pair of intersecting rays received into the eye may be 

 regarded as producing interference-results, yet only such need 

 be considered as have their effect neutralized in the least 

 degree by that of other pairs of interfering rays. 



Since, in every case of experiment, the incident light is not 

 a simple plane wave, but consists of many plane waves, only 

 limited in direction by the size of the pupil of the eye or 

 object-glass of the observing-instrument, and therefore in 

 general but little inclined to one another, we shall only be 

 able to observe those colour-effects in which these various 

 waves combine to the most complete extent. 



