On the Cause of Elliptical Polarization, 1 1 



assumed that the molecules of the undulating medium are, in 

 their state of equilibrium, so arranged in pairs as to make the 

 sums composed of odd powers of differences vanish. When 

 this is not the case the phaenomenon of elliptical polarization 

 is the result. 



We have found (see L. and E. Phil. Mag., vol. ix. p. 421.) 

 that when the displacements f are neglected, 





1^ (r) + .Kr)(A3/ At, -f AijjA?) Aj/J., 

 /<^(r)+4/(^-) (A;yA*, + A^A?) A A. 



(1.) 

 (2.) 



Assume ij = a sin («/— ^o:), 



5 = ;>« sin (nt—kx-^b) ; 

 then 

 A )) = a sin {n t — k x — k A jf) — a sin{w^ — ^ jr) 



= fl5(cos/i: Ax— 1) sin {nt^kx) — as'mk^x cos {nt—kx)-. 

 also, 



A), = a sin {nt—kx — h—k£^x-\-b)'-a sin {nt—k x—b^b) 

 = a {cos(^A^ — 6)-— cos^} sin {nt^kx^b) 

 — a {sin (^Aar— 6)-f sinij. co%{nt — kx—b). 

 In like manner we find, 

 A? = pflsin {nt—kx—b — k ^x)—f)a sm{nt—kx — b) 

 = pa (cos^ Ax — l) sin [ni—kx^b)— p a sin k i^xcos 

 [nt—kx—b) : 

 also, 



A? = pa {cos(^ Aa: + Z>)— cos/;} sin (;?/—^.r) 



— pa {sin {k ^x-\-b)—smb^ cos {nt—kx). 



Now put 



m 5'. {4)(r)+^Kr) Aj/«} (cos ^Ax— 1) = 5, 



7W X. {4> (r) + ^Kr) A2^} (cos^A.r-1) = 5', 



m S. {^{r) + ^{r)Af} sin^Ao- = s^, 



mS,{<t>{r)+^(r)Az''}sinkAx=:s\, 



m H ,^ (r) At/ Az { cos (k A x — b) — cos b] — s^y 



VI X.^ (?•) At/ A z { cos (k A X -{-b) — cos b} = s'^ , 



7n X .^ (r) Aj/ Am {sin (A: A j:— />) + sin />} = ^3, 



mS ,^ (r) A 2/ A 2 {sin (A: A ,2? + ^)-- sin ^} = s'.^; 



and substitute, in the equations (1.), the values of jj, ?, Arj, A ? ; 



C2 



