Mr. Tovey on the Ahsorptioii of Light. 451 



Since we suppose x to be normal to the wave surface, and, 

 consequently, the general expressions for the displacements 

 ij and ^ to be functions of x and /, we may conceive each of 

 these expressions to be developed in a series of the powers eind 

 products of e*' and e^: where e is the number whose hyper- 

 bolical logarithm is unity. Every terrn of the developments 

 will be of the form « e'' ^ + "^ ■* : where a, v, x, are constant quan- 

 tities. 



Suppose ») = «(?^^+'^^, ? = pa<?^^+'^"*', (4.) 



and put e*' ^ + '^ *' = w : (5.) 



we shall then find 



A,j = (e'=^*- 1) «co, A? = (e''^^- l)^aw. (7.) 



Put SjaCe*^^^- 1) = 5, 



tj>' {e''^^ ^\)=s\ (8.) 



and substitute the values (6.) and (7.) in (3.); we shall then 

 have, on omitting the common factor « w, 



V^ = 5 + p S^ , 



p 



Hence it appears that, when the arbitrary quantities v, p, x, 

 are made to satisfy these two conditions, the expressions (4<.) 

 are a particular solution of the equations (3.). Now, since 

 these equations are of the first degree, they may be satisfied, 

 not only by the expressions (4.), but by the sums of any num- 

 ber of expressions of the same form ; therefore we may put 



>j = 2«e'^^ + ''^, ^= Sp«e^^ + '^'^; (10.) 



which will represent the general values of >] and ^, developed, 

 as we have previously supposed, in series of the powers and 

 products of ^ and e^. 



It is obvious, by (10.), that the values of v must be entirely 

 imaginary; for, if not, the displacements, >] and ^, would con- 

 tain terms increasing indefinitely with the time; which they 

 cannot do, because the motions under consideration are, by 

 hypothesis, merely vibratory. We may therefore put 



v = «>v/^, (11.) 



2 G 2 



