204 DOUBLE REFRACTION AND CIRCULAR POLARIZATION 



which has a very simple geometrical signification. If we consider 

 the ellipsoid ax*+by* + cz*+eyz+fzx+gxy, (24) 



and especially its central section by a plane parallel to the planes of 

 the wave-system which we are considering, it will easily appear that 

 the equation 



2 ) = 



will hold of any two points x lt y-^, % and x 2 , y 2 , z z which belong to 

 conjugate diameters of this central section. Therefore equation (23) 

 expresses that the displacements a lt /3 l} y x and a 2 , /3 2 , y 2 are parallel 

 to conjugate diameters of the central section of the ellipsoid (24) by a 

 wave-plane. But since the displacements a 1? /3 lf y x and a 2 , /3 2 , y 2 are 

 also at right angles to each other, it follows that they are parallel 

 to the axes of the central section of the ellipsoid (24) by a wave-plane. 

 That is: The axes of the displacement-ellipse coincide in direction 

 with those of a central section of the ellipsoid (24) by a wave-plane. 



16. If we write Uj, U 2 for the reciprocals of the semi-axes of the 

 central section of the ellipsoid (24) by a wave-plane, Uj being the 

 reciprocal of the one to which the displacement a lt /3 1? y x is parallel, 

 we have 



), (25) 



as is at once evident if we substitute the coordinates of an extremity 

 of the axis for the proportional quantities a v /3 V y r So also 



aa 2 HW + ^y2 2 + ^ 2 y 2 +/y 2 a 2 +^a 2 ^ 2 = U 2 2 (a 2 2 + ft 2 + y 2 J ). (26) 



If we write V for the velocity of propagation of the system of 

 progressive waves corresponding to the system of stationary waves 

 which we have been considering, we shall have 



V = l (27) 



P 



By equations (22), (25), and (26), equations (18) and (20) are 

 reduced to the form 



TWi^ftA-VW, U 2 W| AP2 =VW> (28) 



where we are to read + or according as the disturbance has the 

 character of a right-handed or a left-handed screw. In a progressive 

 system of waves, when the combination of displacements has the 

 character of a right-handed screw, the rotations will be such as appear 

 clockwise to the observer, who looks in the direction opposite to that 

 of the propagation of light. We shall call such a ray right-handed. 

 We may here observe that in case = the solution of these 





