22 Prof. Airy on the Phenomena of Newton's Rings, as formed 



J co s» / tanji — «')\ 

 ^""cos7V ~tan(i + »')/ 



rn«./ tan(»-/)\ 



tan (*' - «") 

 tan (i' + *", 

 tan (i r — ») 



8 ~ tan (i' + M 



h = 



tan (V + *)' 



a J. COSi ' / _ tan (/ — 1) \ 

 ~"cosT\ tan (7 + *)/ 



Hence fk = 1 — e*, and g /z = — g e ; and the expression 

 becomes 



a e sin -- (v t — x) ■{- ag(l —e 1 ). 



A 



sin (— (vt — x) — V\ +ge . sin f — (wtf — jr) J 



1 -f2gecos V+g*e~ 

 Resolving this into the form 



Psin — (vt — x) -f Qcos -— (vt~ x), 



A A 



the intensity or P 2 -f Q 9 becomes 



2 £ 2 + e°~ + 2 £ e cos F 

 - 1 +2ge cos F + g 2 ^* 



The maxima and minima of this correspond to the maxima 



and minima, or the contrary, of cos V, When V—0 9 2 w, &c. 



a 2 X 

 that is when T— 0, or = „ or = 5 » &c. the inten- 



2 cos i" 



sity of the reflected light is 





and when T= -, , &c. the intensity is 



4- cos i 4 cos i J 



'(£*)■ 



and the excess of the latter above the former is 



» g g(l- g »)(l-^) ^J. 



This is the difference of intensity of the brightest and of the 

 darkest parts of the rings : and when it is positive, the centre 

 of the rings is dark. 



Now tan 2 (j 4- »') is always greater than tan ? (i— <'), and 



