IN PERFECTLY TRANSPARENT MEDIA. 203 



If we differentiate with respect to a lt &, y 1? and write a 2 , /8 2 , y 2 for 

 da v d/3 l} dy lt we obtain 



072 



If we differentiate with respect to 03, j8 2 , y 2 , and write Oj, /3 2 , y 2 for 

 da 2 , cZ^ 2 , dy 2 , we obtain 



aa 2 2 + 6)8 2 8 + cy 2 2 + e 2 y 2 +/y 2 2 + ga 



. (20) 



The equation derived by differentiating with respect to a 2 , /8 2 , y 2 , and 

 writing a 1? /3 lt y for ca 2 , cZ/3 2 , cZy 2 , is identical with (19). We should 

 also observe that equations (18) and (20) by addition give equation 

 (16), which therefore will not need to be considered in addition to 

 the last three equations. 



14. The geometrical signification of our equations may now be 

 simplified by a suitable choice of the position of the origin of 

 coordinates, which is as yet wholly arbitrary. 



We shall hereafter suppose that the origin is placed in a plane of 

 maximum or minimum displacement,* if such there are, In the case 

 of circular polarization, in which the displacements are everywhere 

 equal, its position is immaterial. The lines p 1 and /> 2 , of which a lf /3 lf 

 y l and a 2 , /3 2 , y 2 are respectively the components, will now be the 

 semi-axes of the displacement-ellipse, and therefore at right angles. 

 (See 9.) The case of circular polarization will not constitute any 

 exception. Hence, 



a 1 a 2 + ) 8 1 /3 2 + y 1 y 2 = 0, (21) 



and by 9, 



(22) 



where we are to read + or in the last member according as the 

 system of displacements has the character of a right-handed or a 

 left-handed screw. 



15. Equation (19) is now reduced to the form 



i a 2 + 26/8 A + 2c yi y 2 + e(&y 2 + 7l /3 2 ) 



(23) 



* The reader will perceive that an earlier limitation of the position of the origin by a 

 supposition of this nature, involving a limitation of the values of a 1 , /3j , y lt a 2 , /3 2 , -y 2 , 

 would have been embarrassing in the operations of the last paragraph. 



