156 On a Dynamical Theory of 



of the second degree, containing, in its general form, the squares 

 and products of X, Y, Z, with six constant coefficients. 



Of these coefficients, the three which multiply the products 

 of the variables may always be made to vanish by changing the 

 directions of the axes of #, y, z. For this is a known property 

 of functions of the second degree, when the co-ordinates are the 

 variables ; and we have shown, in Lemma II., that the quan- 

 tities X, Y, Z are transformed by the very same relations as 

 the co-ordinates themselves. Therefore, in every crystal there 

 exist three rectangular axes, with respect to which the function 

 V contains only the squares of X, Y, Z ; and as it will presently 

 appear that the coefficients of the squares must all be negative, 

 in order that the velocity of propagation may never become ima- 

 ginary, we may consequently write, with reference to these axes, 

 V = - \ (a 2 X* + V Y- + c*Z 2 ), (2) 



omitting the constant K as having no effect upon the motion. 



The axes of co-ordinates, in this position, are theprincipal axes 

 of the crystal, and are commonly known by the name of axes of 

 elasticity. Thus the existence of these axes is proved without 

 any hypothesis respecting the arrangement of the particles of 

 the medium. The constants , b, c are the three principal velo- 

 cities of propagation, as we shall see in. the next section. 



Having arrived at the value of Y, we may now take it for 

 the starting point of our theory, and dismiss the assumptions by 

 which we were conducted to it. Supposing, therefore, in the 

 first place, that a plane wave passes through a crystal, we shall 

 seek the laws of its motion from equations (1) and (2), which con- 

 tain everything that is necessary for the solution of the problem. 

 The laws of propagation, as they are called, will in this way be 

 deduced, and they will be found to agree exactly, so far as mag- 

 nitudes are concerned, with those discovered by Fresnel ; but the 

 direction of the vibrations in a polarized ray will be different 

 from that assigned by him. In the second place, we shall in- 

 vestigate the conditions which are fulfilled when light passes 

 out of one medium into another, and we shall thus obtain the 

 laws of reflexion and refraction at the surface of a crystal. 



