By a series of transformations the integral for the radiated energy in the general case is brought into a form which facilitates its evaluation. Two results follow from this. First, it is seen that a generalized uniaxiality condition can be formulated even when the electric and the magnetic anisotropy axes do not coincide. Secondly, a theorem is established which enables one to deduce the results for double anisotropy from those for anisotropy in E only by means of certain simple substitutions.
1. REMARKS ON THE GENERAL CASE
In Part I, closed expressions have been derived for the energy radiated by a charge moving along a principal dielectric axis of a biaxial crystal anisotropic in E only. For the calculation of the r.e. in the general case it proves advantageous to pass on to a frame in which h is diagonal. If the electric and the magnetic anisotropy axes coincide then E becomes diagonal simultaneously with X. After an appropriate change of scale and a redefinition of the dielectric tensor the determinant A(k) of I, Section 2 can then be written in the standard Fresnel form, while the cofactor [all] becomes a linear combination of the cofactors [solo]' of the accented coordinate system in which h' = 1 and E' occurs in place of E. These operations bring the Fourier transform CD(k) of the potential into the 'normal form' in which the h's occur explicitly only in the coefficients of the linear combination.
The reduction to the 'normal form' facilitates evaluation of the integral for the r.e. If the integration over k, is carried out first it is seen, as in the uniaxial case, that the principal values of the integrals for positive and negative k, cancel one another. The contributions to the integral, thus, come only from the poles occurring at the zeros of the denominator of @P(k), that is, only from the points lying on the Fresnel wave surface, f(E', k') = 0. In the expression for the residue it is, therefore, possible to replace kl', k2', k,' by the parameters p, q of the wave surface. The residue has a simple form with [a,,] in the numerator and a linear combination of the partial derivatives 8f /ski' in the denominator.
The next integration over k, has to be carried out on the line along which the 428 Copyright