  Let $\Gamma$ denote a bipartite distance-regular graph with diameter $D \ge
4$ and valency $k \ge 3$. Let $X$ denote the vertex set of $\Gamma$, and let
$A$ denote the adjacency matrix of $\Gamma$. For $x \in X$ let $T=T(x)$ denote
the subalgebra of Mat$_X(\mathbb{C}$ generated by $A, E*_0, E*s_1, \ldots,
E*_D$, where for $0 \le i \le D$, $E*_i$ represents the projection onto the
$i$th subconstituent of $\Gamma$ with respect to $x$. We refer to $T$ as the
{\em Terwilliger algebra} of $\Gamma$ with respect to $x$. An irreducible
$T$-module $W$ is said to be {\em thin} whenever dim $E*_i W \le 1$ for $0 \le
i \le D$. By the {\em endpoint} of $W$ we mean min$\{i | E*_iW \ne 0\}$. For $0
\le i \le D$, let $\Gamma_i(z)$ denote the set of vertices in $X$ that are
distance $i$ from vertex $z$. Define a parameter $\Delta_2$ in terms of the
intersection numbers by $\Delta_2 = (k-2)(c_3-1)-(c_2-1)p^2_{22}$. In this
paper we prove the following are equivalent: (i) $\Delta_2>0$ and for $2 \le i
\le D - 2$ there exist complex scalars $\alpha_i, \beta_i$ with the following
property: for all $x, y, z \in X$ such that $\partial(x, y) = 2, \: \partial(x,
z) = i, \: \partial(y, z) = i$ we have $ \alpha_i + \beta_i |\Gamma_1(x) \cap
\Gamma_1(y) \cap \Gamma_{i-1}(z)| = |\Gamma_{i-1}(x) \cap \Gamma_{i-1}(y) \cap
\Gamma_1(z)|;$ (ii) For all $x \in X$ there exist up to isomorphism exactly two
irreducible modules for the Terwilliger algebra $T(x)$ with endpoint two, and
these modules are thin.
