THEORY OF THE DISPERSION OF LIGHT. 255 



(a) the amplitude of vibration ; {t) the time from the beginning of the disturbance ; 

 (x) the distance along the ray from the molecule first agitated ; while { -r-) is the ve- 

 locity of propagation, which further involves the relation 



7 = T andA = V^• ......... m 



where (jja) is the refractive index, and (K) the wave-length in the medium. 



(6.) The investigations of Fresnel require us to suppose the vibrations performed 

 in planes at right angles to the direction of the ray: or, in general, referring the 

 whole motions to three rectangular axes x, 3/, z, of which we may suppose x to 

 coincide with the direction of the ray, and naming the displacements in each of those 

 directions respectively |, rj, ^, we must suppose 1 = 0, while each of the others are 

 functions of x and t of the form above, viz. 



71 =z'^ {asm (nt — kx)}] 



^^ , ^'l ......... . (3.) 



^ = l{(^sm{nt — kx)}\ 



In polarized light the amplitudes a |3 are in Fresnel's notation 



a = cos i and /3 = sin i, 



where i is the angle formed by the direction of the vibration with that of polarization; 



wlience also 



a2 4.^2_ 1, / (4) 



In plane polarized light either i =z 0, or i = -^ ; hence one of the expressions (3.) 



disappears. 



For elliptically polarized light both are retained ; but one vibration is retarded by 

 a quantity (b), or we must take 



1] = "2 {u sin (n t — k x)} 1 



^ ^^ I (5.) 



Z=^{psm(nt-kx-i-b)}, \ ' 



expressions which will easily be found to give, on substitution of the value of n in that . 

 of ^, the equation to the ellipse described by the etherial molecules, which becomes 



that of a circle if 6 = -^ and a = j3. If /S = the formulas are reduced to those for 



common light, and a and j3 are not restricted by the condition (4.). 



(7.) From (2.) we have ^ = ^X; (6.) 



hence we may observe ; 1st. If -^ be constant there is no dispersion. In this .case it 



is easily found that the formulas (3.) are solutions of the well-known equation of vi- 

 bratory motion, 



d} u n^ (P u 



