256 



PROFESSOR POWELL'S REMARKS ON THE 



2ndly. If n be an independent constant these expressions give the index in the simple 

 inverse ratio of the wave-length, which is manifestly not the law of the unequal refran- 

 gibility of the primary rays, as appears at once from what is termed the irrationality 

 of dispersion. But the application of the formulas to the ordinary phenomena is in- 

 dependent of this, and will be equally valid if ^ be dependent on X, precisely or very 

 nearly in the relation expressed by the formula which has been compared with ex- 

 periment in my previous papers, and which thus explains the unequal refrangibility. 



Now it has been shown by the distinguished mathematicians already mentioned, 

 that, under certain conditions, the expressions (3.) can be derived from equations of 

 motion for the vibrations of an elastic medium on dynamical principles, which also 

 involve the desired relation between jm, and X. It will be necessary to state and ex- 

 plain these equations, which may be most shortly done as follows : 



(8.) Let the coordinates in space of any molecules m m' be respectively 



' (7.) 



m 



X 



m' .r-|-Aa? y -{• ^y ^-j-AzI 



and the distance between them 



r= v/(A^2 + A3/2 + A22) (8.) 



Let the force which maintains the system of molecules as an elastic medium be any 

 function of the distance as / (r) : then it will be easily seen that we have for the 

 forces in the direction of the three axes, in equilibrium, 



2/(r)^" = 



2/(r)^ = 0> 



(9.) 



Let the system be disturbed, and after a time (t) let the displacements in the di- 

 rection of the three axes be respectively 



...I » z ] 



m 



(10.) 



m' ?-i-A| ^-\-/^n ^ + Aq 



also the distance becomes 



r-}-Ar= i/{(Aj:H-A|)2-|-(A3/-hA?7)2-f (A^ + A02| . . . . (n.) 



In this condition it will be easily seen that we have for the forces in the direction of 

 the three axes. 





(12.) 



