(22.) 



Absorption of Light. 453 



0-2 = B /i cos i + B^ h' cos 2 2 + Bg F cos 3 ? + &c. 

 0-3= B 7i sin z + B^ h"- sin 2 / + B2 hP' sin 3 ? + &c. 

 we find 



5^ = 0-2 + -/ - I . CTg . ^^"•'' 



Now suppose 



p = ^ (cos y 4- i/ — 1 . sin y) , (21.) 



and substitute the values of s, 5', s^, and p, in (12.); then, 

 since w is entirely real, we shall have 



7^^ + (T + |3 (o-g cos y — 0-3 sin y) = , 

 ?«^ + cr' + -^(<^2 cos y 4- cTgsin y) = , 

 o-, + /3 (0-2 sin y + (Tg cos y) = , 



< - ^ (0-2 siny - o^scos y)= . 



The four equations (22.) determine the values of//, z, /3, y, 

 corresponding to any given value of n ; and it may easily be 

 seen that these equations will not be affected if we change the 

 sign of w, or if we change, simultaneously, the signs of/ and 

 y. Therefore, since the general values of >) and ^ are the 

 sums of all their particular values, we may give to v, x, p, in 

 (10.), not only the values (11.), (14.), (21.), but also those 

 which result from them, by changing the signs of n, /, y. If 

 we do so, and, for the sake of abridgement, put 



h cos 2 = £ , /i sin z = ^ , (23.) 



/3 (cos y + s/ - 1 . sin y) = pi , 



(24<.) 



/3 (cos y — */ — 1 . sin y) = P2 J 

 we shall have 



ij = S {e^"^ («^e('*^+^^)V^ + «2 e-("^+*'^)V^)} , 



^ = 2{e'^^(p,«,^('*^+^'^)v-i + P2«2^~^"^+*^^'^-^)}; 



where a, ag, are values of a, and are entirely arbitrary. 



By changing the last expressions into circular functions, we 

 find 



>3 = 2 |« c^^' sin {nt + kx + b)] , 

 ? = 2 {a'<7^*' sin {nt-^ lex -{■ V)) : ^ -^ 



where a, Z*, a\ b', are determined by the equations 



a sin 6 = «^ 4- a,^ » ^ cos b = (cc^ — u.^ V' — 1 , 

 a' sin 6' = p^ «/ + f2 «*2 > '*' cos b' = (p^ «/ ~-p2 *i} V --1 . 



