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 formulae 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, 



— =^(S-|)'+KS-I)'+<|4:) ! - « 



it turns out that a, b, 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 §5, Srj, S£. 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. 



