30 Mb. O'BRIEN ON THE ABSORPTION OF LIGHT. 



Also (13) + (14) gives 



2a=(i+^) «'+(l + 7<) »" 07). 



And from (l6) and (17) we have 



I V I ■o\v" v'^ -mB ■»'" - m,B\ 



Now if we suppose the roots of (l) to be impossible, we have 



- = « + >)%/- 1» — = f-»)V -1. 



Making these substitutions in (18), we evidently find an expression for a of the form 



a =/(cos w + \/^l sin to) a, or, fe"^' ■ a; 



where / and o) are real quantities, the former of which does not change sign when - >; is put for n, 



but the latter does. Now a' becomes a" when v" is put in place of v', and v' in place of v" ; i.e. 



when - >; is put for r/. Hence we have 



a" =/e-"V-i . „. 



Hence the general expression for 1^+ V is 



fa {^^-\ e"('-^)v'-'+ e-'-'^KA'-^)^-'] ; 



or, fa {e-'^e I »('--•' + " 1 V-i +e-«''--. el ""-"'-" I V:^} (19). 



(19) therefore is the symbolical disturbance in the upper medium arising from the symbolical 

 disturbance a e"('~u)^/^ in the lower. By changing the signs of n and t], and therefore of u), we 

 find that the symbolical disturbance a e~"('"j)^' in the lower medium gives rise, in the upper, to 



fa {e'^'.e- i "<'-"'+" 1 V^ + e'"'''. e~ ! "«-"'-" 1 V-i| . 

 Hence, superposing these two sets of disturbances, we find that the real disturbance 



H) 



in the lower medium gives rise to the real disturbance 



fa [f-. cos {w(< - e ;:;) + (o} + 6""'^ cos {n(t - e as) - co]] 

 in the upper. 



Now this latter expression indicates a continually increasing intensity, and therefore if the roots of 

 (1) were impossible, light after refraction would continually increase in intensity in passing through 

 the refracting substance ; a result which is quite at variance with experiment. Hence we may 

 conclude that the roots of (1) cannot be impossible, and that the explanation of absorption given 

 above is not true. In fact, that explanation falls to the ground if we be not at liberty to reject the 

 integral . a e"''' cos n(t -ex) and retain a e""""^ cos n(t - e x), which we cannot do without violating 

 the equations of connection, as is evident from the process just gone through. 



It appears, therefore, that though the action of the material upon the ethereal particles afibrds a 

 complete and satisfactory explanation of dispersion, we must look to some other source for an 

 explanation of absorption. 



