on the Undulatory Hypothesis of Light, 473 



dl) 



But -j may be taken with sufficient approximation to be the 



actual accelerative force of the atom. Consequently 

 2 d8__e 2 d 2 x 

 ~~ e 'd^~~¥'~a¥' 

 Thus the total accelerative force of the atom is 



( &tt dx\ e d x 



d 2 x 

 Equating this to —j-^, we have for determining the motion, 



d 2 x k dx km . 2ir ,, , 



b 2 b 2 



In this equation we may neglect in the circular function the 

 small variation of x about its mean value, and suppose the mean 

 value to be the arbitrary quantity c. Then putting, for the sake 



of shortness, k = qll — j^f, we have to integrate the equation 

 d q x dx . 2irbi 



I may here take occasion to state that so long ago as the year 

 1830 1 attempted to ascertain the motions of an atom of a medium, 

 resulting from the action of a series of undulations, upon prin- 

 ciples identical with those adopted above, with the exception thaT* 

 the action of the elasticity of the medium was supposed to depend 

 on the absolute displacement of the atom instead of the relative 

 displacement of adjacent atoms. (See the Philosophical Maga- 

 zine for September 1830, p. 169.) In consequence of that sup- 

 position the investigation did not altogether succeed in account- 

 ing for the different refrangibility of the rays of light, which was 

 its ultimate object. It will presently appear that that problem 

 may be satisfactorily solved by the integration of the foregoing 

 differential equation. 



By the integration of the equation, and putting cot <p for — — , 



we obtain 



dx . /2irbt , \ n - q t 



— -=s — W2sm0eos( — - — j-91-f-Ls , 



from which equation the term containing the arbitrary constant 

 will soon disappear. Hence 



2irbt dx , . fiirbt \ 



-j- = m cos 9 sin I — — -j- <p ). 



msm 



Phil. Mag. S. 4. Vol. 26. No, 177. Dec. 1863. 2 1 



