230 
Boyal Irish Academy. 
entirely in the surface of the wave. From this property he shows, 
in the first place, that v is a function of the three differences 
dr) d C d^ d^ d'E, dr) 
dz dy dx dz dy dx’ 
and, in the next place, that the only part of it which comes into 
play is of the second order, containing the squares and products of 
those quantities, with of course six constant coefficients. Then, 
supposing the axes of coordinates to be changed, he proves that 
the usual formulae for the transformation of coordinates apply also 
to the transformation of those differences ; so that, by assuming the 
new axes properly, the terms in the function v which depend on the 
products of the differences may be made to vanish, and v will then 
contain only the three squares, each multiplied by a constant co- 
efi&cient. The axes of coordinates in this position are defined to 
be the principal axes, (commonly called the axes of elasticity) ; and 
when we put, with reference to these axes. 
it turns out that a, h, c, are the three principal velocities of propaga- 
tion within the crystal. 
To find the laws of propagation in a continuous medium of inde- 
finite extent, we have only to take the variation of v from the ex- 
pression (2), and, after substituting it in the right-hand member of 
equation (1), to integrate by parts, so as to get rid of the differential 
coefficients of the variations SE, St), Then equating the quan- 
tities by which these variations are respectively multiplied in the 
triple integrals on each side of the equation, we obtain the value of 
the force acting on each particle in directions parallel to the principal 
axes. The double integrals which remain on the right-hand side of 
the equation are to be neglected, as they belong to the limits which 
are infinitely distant. The resolved values of the force thus obtained 
lead to the precise laws of double refraction which were discovered 
by Fresnel, with this difference only, that the vibrations come out 
to be parallel to the plane of polarization, whereas he supposed them 
to be perpendicular to it. 
When there are two contiguous media, and the light passes out 
of one into the other, suppose out of an ordinary into an extraor- 
dinary one, and we wish to determine the laws of the reflected and 
refracted vibrations, it is only necessary to attend to the double in- 
tegrals in the equation of limits ; but the integrations must now be 
performed with respect to other coordinates. Taking the separa- 
ting surface of the two media for the new plane of xy, the axis of x 
being in the plane of incidence, let the principal axis x of the crystal 
make with these new axes the angles a, /3, y, while the principal 
axes y and 2 ;, in like manner, make with them the angles a', /3', y', 
and a", ft", y", respectively. Then, marking with accents the quan- 
tities relative to the new coordinates, we have 
