Prof. Airy on Newton's Rings. 21 



reflected by the quantity corresponding to the space 2 Tdos /: 

 or if — (vt — x) be still taken as the measure of the phase of 



A 



the ray first reflected, — (vt—x) — Tcos i' will be that of 



A A 



the ray which has been reflected at the surface of the second 

 medium and then enters the first. The quantity — - T cos »' 



we shall for abbreviation call V. Of the light which reaches 

 the surface of the first medium, a part will be partially re- 

 flected at the surface of the second medium, and will partially 



enter the first medium : its phase will be -— (vt — x) — 2 V; 



A 



and so for succeeding reflexions. 



Now suppose that at the last surface of the first medium, 

 the coefficient of the incident vibration being 1, that of the re- 

 flected vibration is e, and that of the refracted^/*; at the first 

 surface of the second medium, suppose the coefficient of the 

 reflected vibration to beg; and for light incident from air on 

 the surface of the first medium, suppose the coefficients of the 

 reflected and refracted vibrations to be h and k. Then, the 

 coefficient in the incident light being a, 



That in the first reflected light is ae 



that in the refracted light is af 



that in the light reflected at the second medium is afg 

 and that in the light refracted into the first medium is ...afgk 

 that in the light reflected from the first medium is afg h 

 that in the light reflected from the second medium is afg'h 

 and that in the light refracted into the first medium is... afg^hk 

 and so on ; the coefficients after the first following a geome- 

 trical progression whose ratio is g h. Thus it appears that the 



whole vibration will be a . e . sin — (vt — x) + a .fg k 



A 



fsm(^~(vt-x)-F\+gh.sin( — (vt-x)-<2V\ +&c.\, 

 or a . c . sin -— - (v t — x) + a .fg k . 



A 



s\n(-~(vt — x) — V\ — gh.sm ( ~ (vt — x)\ 



~l - 2gJi^coW-[^h i ~ 

 Now in FresnePs expressions, 

 tan (i — i') 



£ = 



tan (t + *')' 



