181' Researches in the Undulatory Theory of Light. 



intensity is proportional io a^ e '^-f ^^ a" e *^* : hence, if we 



put c = fl^+ jS^ a^, it will be proportional \o ce ^^. Sup- 

 pose the origin of x to be at the surface of any medium on 



which the light falls ; then c e ^'*' will be the intensity of the 

 light after it has traversed a thickness of the medium equal 

 to iv. And if Cy, Cg, Cg, ... Sy, e.2> ^2' • • • ^® ^^ values of 

 (a-+ /S^a-) and s respectively in the general expressions (35.), 

 the intensity of the light in the transmitted ray will be 



c^e ' + Cc,e ^ + c^e ^ + &c. (46.) 



If we were to put y, y\7/', ... for e ^' , e ^^ , e ^% . . . in 

 this formula, it would become the very same as that which 

 was devised by Sir John Herschel to represent the law of 

 absorption as indicated by experiments. 



The formula (46.) shows clearly enough the manner in 

 which the absorption depends on the thickness of the medium, 

 and it indicates, by the different values of e, which belong 

 respectively to different values of /r, that the absorption is 

 different for waves of different lengths. But the relation of 

 s to k, which, as we have seen, is implicitly expressed by the 

 equation (42.), is so extremely complicated that the readiest 

 way of testing our theory with reference to it, seems to be 

 by inquiring whether experiments show that it is of so com- 

 plex a character. iVow the nature of this relation, as in- 

 ferred from experiments, is stated by Sir J. Herschel in his 

 excellent paper on the absorption of light, published in vol. iii. 

 of the current series of your Journal, where at page 402, he 

 says: — " If we represent the total intensity of the light, in 

 any point of a partially absorbed spectrum, by the ordinate 

 of a curve, whose abscissa indicates the place of the ray in 

 the order of refrangibility, it will be evident from the enor- 

 mous number of maxima and minima it admits, and from 

 the sudden starts and frequent annihilations of its value 

 through a considerable amplitude of its abscissa, that its equa- 

 tion, if reducible at all to analytical expression, must be of 

 a singular and complex nature, and must at all events involve 

 a great number of arbitrary constants, dependent on the re- 

 lation of the medium to light, as well as transcendents of a 

 high and intricate order." This character is very suitable 

 to our equation (42.), and may, therefore, be taken as an evi- 

 dence of its truth. 



That a spectrum absorbed in the apparently capricious 

 manner described in the above extract, would result from the 

 relation between e and k implied in (42.) may be thus shown. 

 Suppose, in the first place, s to be zero, and k^^k^y k^^ ... 



