  We present analytic analyses for the joint density of states (JDOS) and the
elastic scattering susceptibility (ESS) for d-wave superconductors under the
linear dispersion approximation. The profile of the JDOS, determined by the
curves on which the JDOS diverges, is found to be the envelope function related
to the contour of constant energy of the Bogoliubov quasiparticles. We derive
the explicit form of the envelope functions. The intrinsic relationship of the
JDOS patten and the octet model of ESS is clearly revealed. It is also found
that apexes of the envelope function coincide with the octet vectors when the
Dirac-cone anisotropy is strong, which is relevant to the d-wave cuprates.
