AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 



19 



since it would contain no disposable quantity, p^, p', p", s.^, «', s", being all given in terms 

 of p by (5). We have therefore Sj = - s. 



For the reasons given in Article (l6), we must take the positive values of / and s", and 

 the negative value s^ ; i. e. we must put 



s'=+\/l-^-^, a"= + V 1 _ (^^ja'j (10), s,= -\/l-{^pY (11). 



Hence, if we take the arrows A'', Ni, N^, N', N", 

 to mark the directions of the rays, as before. Fig. (3) 

 will represent the circumstances of the case, and we 

 have 



t N^AZ'= /. NAZ, sin N'AZ = m sin NAZ', 



sin NiAZ' = — sin NAZ', sin N"AZ = — sin NAZ'. 



V V 



Thus, both the reflected and the refracted normal 



// 

 . . "a , " . 



ray obey the law of refraction, putting — and — in- 



V V 



stead of /u. The transversal rays are circumstanced 

 just as in the preceding case. 



(20.) We now proceed to compare the intensities of the rays N, JV,, and N', and we 

 shall do this, first, on the hypothesis that normal waves are propagated very slowly compared 

 with transversal. 



On this hypothesis we may suppose that — and — are zero, and then, by (5), we have, 

 P2 = 0, p"=0. Hence, writing a, a,, a', for F, K,, V, as before, we have by (6) and (8), 



a + ai = ixa, s {a — a,) = s'a, 



J ., „ fiS - s' , 2s 

 and therefore aj = j a, a = a. 



fJiS + s 



These are identical with Fresnel's formula for light polarized at right angles to the plane 

 of incidence. 



To determine a.^ and a", we have, by (7) and (9), (observing that 8.^=- 1, s" = l by (lo) 

 and (11) ) 



/' 

 a^ a 



p{a + a,) -a^ = p'a' + a", or a^ + a"= (jip - p) a by (6) ; 



V, fj." - I 2ps „ v" 



therefore a. = 



•0, 



a = — a, 



«2 



/* flS + s 



(21.) We shall now make a different hypothesis, and suppose that v^ is equal or very nearly 

 equal to v". 



On this hypothesis, we have by (5) p^^p", and by (lO) and (11) s^ = - s" ; therefore by 

 (6) and (8) we obtain 



a + a^ = jia , s (o — aj = s a, 



which give us Fresnel's formulas just as before. 



Also by (7) and (9) we have 



"a = a") P (« + fli) - s"c2 = pa' + s"a" ; 



„ fj.' - 1 2ps 



therefore a.^ = «" = ■ 



ixs 



fia + s 



-, .a. 



02 



