376 
_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 formule for the trans- 
formation 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 coefficient. 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, 
dn de o(de dé dé dy 
Tigi la i+ (2-F)+e Gann a) (2) 
it turns out that a, 6, c, are the three principal velocities of 
propagation within the crystal. 
To find the laws of propagation in a continuous medium 
of indefinite extent, we have only to take the variation of 
v from the expression (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 varia- 
tions 0&, dn, 6¢. Then equating the quantities 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 pre- 
cise 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 polarisation, whereas he 
_ supposed them to be perpendicular to it. 
