FresneVs Reflexion of Light IJieory. 317 



_ ^-6^+1+ */Q* 4 -S> 2 -H) 2 - 36/x 4 rq . 



These are values of % that were shown to make f(%) a 

 minimum, and they are arrived at by ascribing to n a value 

 such that p shall vanish, thus furnishing one limit for the 

 use of —y/p and + s/p. 



Again, the ratio of light reflected at normal incidence is 

 0*-l) 2 /(>+l) 2 , and if we write (^ + l) 2 /(/4-l) 2 for n, 

 as n occurs in p and q, then 



V/? = p? + l — S> 2 /i> — I) 2 ; 



and if —y/p be adopted, then 



<Z = °> %=/ A " 1 ; sini = 0. 



This value, (jl + l) 2 /(/x — l) 2 , of ft determines the other 

 limit of ft for which — y/p can be employed ; therefore, these 

 limits show that the negative sign of y/p is applicable to 

 the range of i from normal incidence to the incidence where 

 minimum reflexion occurs. 



Bui, if we make the latter substitution for n in p and q 

 and adopt + \/p, then, 



• 9 = 2[<y-i)*- vo*+ i)70i-i)»]», 



an expression, however, which is real only when pu lies outside 

 the limits 2 + \/3. This is the value of V^ (in conjunc- 

 tion with n = (/x -h 1 ) 2 /(/x — 1 ) 2 throughout the % value)' that 

 is required in order to determine the other angle of incidence 

 where the light reflected is equal to that reflected at normal 

 incidence ; and in (9) we found a limit to the use of the 

 positive sign of y/p at the incidence where minimum reflexion 

 occurs, hence the + sign of y/p is applicable to the range 

 of i from where minimum reflexion occurs to grazino- inci- 

 dence. 



The graphs in figs. 2 and 3 will usefully illustrate these 

 varying relations of y/p to q when jjl > and < 2+^3. 

 Writing 



(2n -?i* + l)(^±l)+2p 2 (2n+n 2 -l) 



