1 92 ■ UNDULATORY THEORY OF LIGHT. 



Combining (his with the equation of living forces given above, and reducing, 

 we obtain these results : 



,., /'taxi('.—p)\^ f 2cos.'sin/) "X^ 

 v/^— I — \ — L' |,andw'-=:i --— ,- — ^, |. [2.3.1 [24.1 



Replacing, as before, the value of sin/^ by its equivalent derived from the 

 equation sin£=:«.shi/', wi; arrive at the following values which embrace only ouy 

 variable : 



-,__ =3 I ; and u^^=\ -^ ~ -~ ■ = | . [25.] 26. 



V u'^ — sm^c + n'^coscy \v n'^ — sm^i+n^coscy 



The following forms are convenient for discussiou: 



1. For vibration in the plane of reflection — 



ncosp — cosi 2ncosi 



V= '—. . H= j- . 



ncoiip-{-cosi ?icosp-j-cost 



2. For vibration normal to the plane of reflection — 



, ncosi — eo.«p , 2mcos( 



v'= 1 -. u'= 1 . 



ncost+cosp Mcosi+cosp 



At a perpendicular incidence cost=:l and cosp=L Hence, in both cases, 



■n— 1 , , 2n 



Thus u and u' \vill always be positive, and v and v' will be positive when 7i is greater than 

 1 and negative when n is less than ]. This ought to bo so, according to the laws of impac* 

 of elastic bodies, because, the density of the ether being by hypothesis the same in both 

 media, the masses leactnig on each other will bo as sint to smp, or as n to 1. 



As the incidence mcrcases, the variations of the value of v and v' will be dissimilar. 

 Whenvib.a ion is in the plane of reflection and 9i exceeds unity, the positive term wcosp 

 is neceasaniy ii'ways gi eater than the negative term cos/.. Both these terms diminish as / 

 increases. Jf they diuunished at the same rate, the value of v Avould be constant. But 

 as I is always gieater than p and neither exceeds 90°, the rate of diminution of cosi is more 

 rapid than iha of cosp, and the value of v increases with the incidence. The same is true 

 when n is less than unity ; only in that case the increasing value of v is negative. When 

 the incidence is maximum, or i^'JL^, n being greater than 1, cosi=0 and v=l; that is to 

 say, the ullecied is equal to the incident light. 



For the amount trausmiLi,cd in the same case, we have, at a perpendicular incidence — 



"""«+! ■ 



And for t=DC.°, or the maxunum incidence, 7i being greater than 1, 



M=0. 



It is also apparent that the amount transmitted constantly diminishes as l increases. 



When vibialion is normal to the plane of reflection, the positive term in the value of v', 

 which is ncosi, Is at fn.-,t gieater (k being greater than unity) or less (w being less than unity) 

 than the r.cgalive term co:-p. But since at the maximum incidence cosi=:U, there must b« 

 some value of i which w'.a give ncost=cosp, or jicosi — co,sp=0. Accordingly, at this inci- 

 dence no light wi;l be reflected. 



The two conditions — 



mcost=cosp, and nsinp=sin£, 

 give immediately — 



sinp=cos;, or <+p=90°. 



The incidence i is the polarizing incidence ; and we here sec that it fulfils the law of 

 Brewster. 



At the maximum incidence, n being greater tlian t, v'= — 1. The sign of the molecular 

 movement, theieibre, changes at the polarizing angle. 



The transmitted light is m this case, at the perpendicular incidence, 



,_ 2w 



And at the maximum incidence, 



u'=0. 



At the polarizing incidence, where, as we have just seen, cosp=Jico8£, 



, 2?? cost 



«'= , =1. 



ncosi-|-ncost 



