30 Mr. Mac Cuttacu on the dynamical Theory of 
and then, if we put 
8 = a’? cos 2a + 6’ cos *B + c* cos *y, (8) 
s? = & cos?a’ +b? cos *B’ + ¢* cos *y’, 
the equations (6) will be reduced to the well-known form 
es as 
de—* de® dé — * dz* (9) 
This result shows that, when the directions of 2’ and 7’ fulfil the condition 
(7), the vibrations é and / are propagated independently of each other, the 
former with the velocity 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 parallel to x’, the other 
parallel to 7/’. 
When the plane of the wave is parallel to one of the principal 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 a, b, or ce, according as the wave is parallel to the axis of z, y, or z. These 
constants are therefore called the principal velocities 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 corres- 
ponding 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 a, but parallel to it; and it is to be 
observed that a difference of the same character distinguishes his views, through- 
out, from the results of the present theory. It will appear in fact, by what im- 
mediately follows, that the equations (7), (8), (9), express exactly the laws of 
Fresnel, provided the quantities & and 7’, in the equations (9), be interchanged. 
To make these laws agree with our theory, it is therefore necessary to alter them in 
one particular, and in one only ; it is necessary to suppose that the direction of 
the vibrations is always perpendicular to that assigned by Fresnel. And since, 
in order to make his views agree with the phenomena, Fresnel was obliged to 
say that, in an ordinary medium, the vibrations of a ray polarised in a certain 
plane are perpendicular to that plane, it is clear that, on the present principles, 
