COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 409 



Wave in film, i.e., between 2 = and z = d, 



E = 0, A' exj, { t p (t - z/V) } + B' exp {ip(t + z/V) | , 0, 

 H = - c/V [A' exp \ip(t- z/V) J - - B' exp \,.p(l + z/V) }], 0, 0. 

 Transmitted wave 



E = 0, (J exp \ip (t - z/c)\, ; H = - exp >i p( t - z/c}} , 0, 0, 



where c/V = n (1 IK). 



The boundary conditions at z = (/ give 



AV- a "'"- /A exp { - t . 27r</H/X} + BV"""' M exp {i . 2*- ^/,/X) = (J exp { - ti> 

 (1 -- IK) [A'c<--*''"< * exp { i. 2ff rfn/X} -- BV"' ' M exp {I-JTT </"/X}] 



= (' exp {i-2-ird/K} 



It follows from these equations that B' is of the order of A/c~ ''"'""*; if therefore 

 ird>iK/\ > 1 we shall be correct within ~2 per cent. (<'"') when we neglect B'. Thus 

 referring to the Table IV. it appears that if a piece of gold leaf before annealing 

 be so thick that d > X/1 '5 or (/ > |X, then, so far as yellow and red light are concerned, 

 TrdnK/\ will be > L for all values of /j.> '5, if we suppose d to vary inversely as p, 

 the number l - 5 being the smallest value of TDIK/IJ. for gold for values of /x from '5 up 

 to unity. 



Eliminating BV"'""* from the last two equations above, 



+ "( l "~ l(C H exp {- i 



n(l IK) J 



From the boundary condition at ~ = we obtain 



Af (I + n (l -)} = 2. 

 Eliminating A' from the last two equations 



ex - l2 ,rfXl_ n = 4** ...... 



Taking the moduli, the ratio | C | 2 of the intensity of the transmitted to that of 

 the incident light is given by 



I p I a _ " ~ K ~ (,-* '<</* (-26) 



- |(1 + n ) 2 +nV} 2 



It appears that when the thickness exceeds f of the wave-length, the absorption is 

 governed by HK ; but, to the same order of approximation, by H~K when d is less 

 than ^-X. 



VOL. CCIII. A. 3 G 



