1902.] 



of FresneVs Polarisation-vector. 



41 



Now the components of the polarisation-vector of a stream of 

 elliptically polarised light may be represented by the real parts of 



u = sD, v = (3D, w = yD, D = A . exp. {ik (Ix + my + nz - at)} , 



bars over the letters representing that they are complex, provided 

 that the ratio a : j3 : y is not real, and if we so choose the origin of time 

 that _ 



ccA = ccL + iaf L', ft A = /SL + t/3'L', yA = yL + ty'L', 



then (a, ft, y), (a, ft, y') are the direction-cosines of the axes of the 

 ellipse traced by the extremity of the polarisation-vector and L, L' 

 are the length of the axes in these directions. 



Taking again equation (1) to represent the ellipsoid of polarisation, 

 we obtain in place of equations (2) the two sets of equations 



FZ 



an - u 2 + 5 -J- j oc + a 12 /S + a 13 y 

 «i 2 a + f a 22 - w 2 + S /3 + «23y = Fm 



and 



fli 3 a + a 2 zp + ^33 ~ 0)2 + ~ 



= Fn 



(8). 



/flu - o> 2 + £ j-,ja' + fli 2 ft + ai 3 y = F7 

 « 12 a' + (a 2 2 - w 2 + ~ k -jijP' + ^237' 

 fl 13 a' + (223/5' + ^33 - + ^ -^7 j y' 



F 



F'wi !, 

 F'ti 



(»>. 



where F is given by (3) and F' obtained from it by writing a', ft, y 

 for a, ft y. 



Whence we have 



(flu - to 2 ) a + <2i2^ + ai 3 y = FZ - (I/a + tLa') p/(icA) ... (10), 



and two similar equations, F being obtained from F by writing a, ft y 

 for a, ft y. 



Now (a, ft y), (a', ft, y'), (I, m, n) being the direction-cosines of 

 three vectors at right-angles to one another, we have 



a! = ym - fin, a = '•- (y'm - fi'n), 



and 



I/a + tLa' = (iLy - L'y') m - (tL/5 - I/ft) n = t (my - ?ift)A, 

 whence (10) becomes 



(fl n - a) 2 ) a + a 12 /3 + fl 13 y = ¥l -i (my - nfi) /d/k (11) 



and two similar equations. 



