Form 0/ Newton's Rings. 243 



Section II. The Shape of the Rings. 



The only difference between this expression and that of 

 equation (viii.) of Sect. II. Part I. is in the 2nd term, 

 a, /3, y taking the place of I, m, n. 



It is evident that the same process as that adopted in the 

 former part of the paper will give similar results. 



Putting 



sin 6=a, 0=/3, cos #=7, 



(0 is the angle of emergence) 

 the equation of the " surface of interference," referred to a 

 system of axes of which the 2-axis is in the direction of 

 emergence, and the substitutions are those of Sect. III. 

 Part I., is 



^+f)sme-2A-2i/ 2 cos 2 6(z + z 1 ) = 0. . . (i.) 

 .-. A k = (x 2 +y 2 ) +fc— ^ a- ^-^tan 6 



r v * yLt COS 6 X T l 



= -(* 2 + z/ 2 )cos0 I"l + - / i-^itan6> 1 ] 

 r ^ L r cos # J 



The other equation to the rings is, therefore, 



^- cos 1 + - 0/ tan 0. = hX or — ^r— X 



r L r cos^fl L J 2 



according as we consider the bright or dark rings. 



Taking the former and putting a 2 for — j. , this becomes 



^+^=« 2 [!-^^4 • • • («■) 



Thus in this case the rings lie on elliptic cylinders which are 

 not co-axial. 



The distance of the centre of the Ath bright ring from 

 the central spot, from the origin, is 



x ^2 \ „2 J 



r cos 2 6 J 2 cos 3 * 



This increases with h and with X, but is of the order Xl unless 



IT 



6 be nearly ^ , when these differences would become very 



perceptible. In ordinary cases the difference is not ob- 

 servable, and the rings are practically co-axial. 



The shape of the surface of interference is exactly the same 

 as that considered in the earlier part of this paper, and the 

 conditions of distinctness remain unaltered. 



December 1889. 



