160 On a Dynamical Theory of 



and then, if we put 



s 2 = a 2 cos 2 a + b* cos 2 /3 + c 1 cos 2 y, 



(8) 

 s' 2 = a 2 cos 2 a' + i 2 .cos 2 /3',!+ c 2 cos 2 7', 



the equations (6) will be reduced to the well-known form 



This result shows that, when the directions of x' and y' fulfil 

 the condition (7), the vibrations ' and i/ are propagated inde- 

 pendently of each other, the former with the velocity of s', the 

 latter with the velocity s. The vibrations must therefore be 

 parallel exclusively to one or other of these directions, else the 

 system of waves will split into two systems, one vibrating pa- 

 rallel to #', the other parallel to y 1 '. 



When the plane of the wave is parallel to one of the prin- 

 cipal axes, it is easy to infer that the vibrations must be either 

 parallel or perpendicular to that axis ; and that, in the latter 

 case, the velocity of propagation is constant, being equal to 

 o, >, or c, according as the wave is parallel to the axis of #, y, 

 or 2. These constants are therefore called the principal velo- 

 cities of propagation ; and we now perceive the reason of the 

 negative sign in equation (2) ; for if any of the terms in the 

 right-hand member of that equation were positive, the corre- 

 sponding velocity would be imaginary. 



According to Fresnel, the wave which is propagated with 

 the velocity a has its vibrations not perpendicular to the axis 

 of x, but parallel to it ; and it is to be observed that a difference 

 of the same character distinguishes his views, throughout, from 

 the results of the present theory. It will appear in fact, by 

 what immediately follows, that the equations (7), (8), (9), ex- 

 press exactly the laws of Fresnel, provided the quantities % and 

 r{, in the equations (9), be interchanged. To make these laws 

 agree with our theory, it is therefore necessary tc them in 

 one particular, and in one only ; it is necessary tr, se that 



