302 ON THE PROPAGATION OF LIGHT 



and represent its three semi-axes by r', r", and r'", the directions 

 of these axes will be the required directions of the disturbance, 

 and the corresponding velocities of propagation will be given by 



Fresnel supposes those vibrations of the particles of the 

 luminiferous ether which affect the eye, to be accurately in the 

 front of the wave. 



Let us therefore investigate the relation which must exist 

 between our coefficients, in order to satisfy this condition for 

 two out of our three waves, the remaining one in consequence 

 being necessarily propagated by normal vibrations. 



For this we may remark, that the equation of a plane 

 parallel to the wave's front is 



= ax' + ly + cz ...... (a) 



If therefore we make 



x = x 



y = y' + l\ } 



z = z + c\, 



and substitute these values in the equation (7) of the ellipsoid ; 

 restoring the values of 



A', ', C', V, E', f, 



the odd powers of X ought to disappear in consequence of the 

 equation (a), whatever may be the position of the wave's front. 

 We thus get 



r = H=I=fjb suppose, 



and P=4-2Z 



which is of the second degree, by changing it, v, and w into x, y, and z. Also 



d d , d . 



^' dy and dz mt a> *> C ' 



This remark will be of use to us afterwards, when we come to consider the 

 most general form of the function due to the internal actions. 



