  This paper presents the properties of a family of scale-free triaxial haloes.
We adduce arguments to suggest that the velocity ellipsoids of such models are
aligned in conical coordinates. We provide an algorithm to find the set of
conically aligned velocity second moments that support a given density against
the gravity field of the halo. The case of the logarithmic ellipsoidal model --
the simplest triaxial generalisation of the familiar isothermal sphere -- is
examined in detail. The velocity dispersions required to hold up the
self-consistent model are analytic. The velocity distribution of the dark
matter can be approximated as a triaxial Gaussian with semiaxes equal to the
velocity dispersions. There are roughly twenty experiments worldwide that are
searching for evidence of scarce interactions between weakly-interacting
massive-particle dark matter (WIMPs) and detector nuclei. The annual modulation
signal, caused by the Earth's rotation around the Sun, is a crucial
discriminant between WIMP events and the background. The greatest rate is in
June, the least in December. We compute the differential detection rate for
energy deposited by the rare WIMP-nucleus interactions in our logarithmic
ellipsoidal halo models. Triaxiality and velocity anisotropy change the total
rate by up to 40 %, and have a substantial effect on the amplitude of the
annual modulation signal. The overall rate is greatest, but the amplitude of
the modulation is weakest, in our radially anisotropic halo models. Even the
sign of the signal can be changed. Restricting attention to low energy events,
the models predict that the maximum rate occurs in December, and not in June.
