316 Mr. R. B. Sangster on some Consequences of 



zero values of/'i'^) will tell us where the turning values of 

 /(%) (obviously minima) occur : — 



„ / /-6^ + l± V Q **--o> 8 +l) 2 -36/A 



These are complex values of %, and it will presently be 

 seen how they are obtained ; meantime, it may be pointed 

 out that they are real values only when the quantity 

 (jj£— 8/^ 2 -f I) 2 — 36/z 4 is positive, which requires /a to lie 

 outside the limits 2x\/3. It should also be noted that they 

 are values of % which have to be written in the sine formula 

 (4) in order to determine the critical angle of incidence 

 where minimum reflexion occurs, and it has been found, on 

 numerical trial, that the + sign of the pair of signs has to 

 be adopted for yit>l, otherwise we are confronted with an 

 unreal sine value. 



We can now determine the proper application of the signs 

 of \/p. It was shown that the two occurrences of ^/p in the 

 value of % must have a similar sign attached for any legitimate 

 value of % so we need only consider the sign of ^/p as it 

 occurs in q. 



Let q be equated to zero, then, n — 1 or f/x+ l) 2 /(//,— I) 2 , 

 according as + or —\/p be adopted. If p be equated 

 to zero, n = or 2(^ 2 -l) 2 /(/x 4 -6/^-f 1). (Incidentally, this 

 limit to the value of n, as it occurs in p. holds good only while 

 n is positive ; whence ^ — 6jjl 2 + 1 = 0, or fi=l± \^2, deter- 

 mines the convergency limits of its application. If [i lie 

 within the limits 1+^/2, y/p is real however great we 

 suppose n to be.) Therefore, in order that^> and q may vanish 

 simultaneously, n must equal both 



0-t-l) 2 /Gu-l) 2 and 2(/* 2 -l) 2 /Ga 4 -6/* 2 + l), 



which can be true only when fi = 2+ V 3, thus making 

 n= (fi+iy 2 j(fji — 1) 2 = 3. These are the values of p, that 

 reflect one-third the incident light at normal incidence, and 

 we have seen that they are turning values in /''(/a -1 ). 



Therefore, when //, lies outside the limits 2+v/3 we need 

 two values of % for values of n greater than (yu,+ l) 2 /(yit— l) 2 , 

 similarly to what was required in the reflexion of light 

 vibrating in the p^ne of incidence, and it is in that case that 

 both signs of ^/p have to be employed. 



When n = 2(/r 2 -l) 2 /(/x 4 -6// 2 + 1), p vanishes, 



s/q ~ ~ ^-by + i 



