  Let $F$ be a non-archimedean local field, of characteristic 0. Let $V$ be a
finite dimensional vector space over $F$ and $q$ be a non-degenerate quadratic
form on $V$. Denote $G$ the special orthogonal group of $(V,q)$. Let $W$ a
non-degenerate hyperplane of $V$, denote $H$ the special orthogonal group of
$W$. Let $\pi$, resp. $\sigma$, an admissible irreducible representation of
$G(F)$, resp. $H(F)$. Denote $m(\sigma,\pi)$ the dimension of the complex space
$Hom_{H(F)}(\pi_{| H(F)},\sigma)$. It's know that $m(\sigma,\pi)=0$ or 1. In a
first paper, we have defined another term $m_{geom}(\sigma,\pi)$. It's an
explicit sum of integrals of functions that can be deduced from the characters
of $\sigma$ and $\pi$. Assume that $\pi$ and $\sigma$ are tempered. Then we
prove the equality $m(\sigma,\pi)=m_{geom}(\sigma,\pi)$. This generalize the
result of the first paper, where $\pi$ was supercuspidal. As in this paper, the
previous equality implies as corollary (assuming certain properties of tempered
$L$-packets) a weak form of the local Gross-Prasad conjecture, now for pairs of
tempered $L$-packets.
