ON THE EEFLEXION AND REFRACTION OF LIGHT. 251 

 du dv dw 



dw dv n dw du du dv 



a = -j- + -j-, P = ^j ^ T~ > 7 = TT"+7T' 

 dy dz' dx dz ' dy dx 



and if the medium is symmetrical with regard to the plane (xy) 

 only, 2 will remain unchanged when z and w are written 

 for z and w. But this alteration evidently changes a and ft to 

 a and ft. Similar observations apply to the planes (xz) 

 (yz). If therefore the medium is merely symmetrical with 

 respect to each of the three co-ordinate planes, we see that 2 

 must remain unaltered when 



or -z, -w, -a, -ft} (z, w, a, ft 



or - y, v, a, 7 > are written for < y, v, a, 7 

 or - x, - u, - ft, - 7 ) la;, u, ft } 7. 



In this way the 21 coefficients are reduced to 9, and the re- 

 sulting function is of the form 



O H + 1 



\dxj \dyj 



n dv dw ~ du dw du dv , . A . 

 + 2P -j-.-j- +2Q-J-.-J- +2R-J-.-J- =^ 2 ... (A}. 

 dy dz dx dz dx dy 



Probably the function just obtained may belong to those 

 crystals which have three axes of elasticity at right angles to 

 each other. 



Suppose now we further restrict the generality of our function 

 by making it symmetrical all round one axis, as that of z for 

 instance. By shifting the axis of x through the infinitely small 

 angle BO, 



becomes \y 



