  Let p be a prime number, and let f, g, and h be three modular forms of
weights $\kappa$, $\lambda$, and $\mu$ for $SL(2,\Bbb{Z})$. We suppose $\kappa
\geq \lambda + \mu$. In joint work with Kudla, one of the authors obtained a
formula for the normalized {\it square root} of the value at $s = {1/2}(\kappa
+ \lambda + \mu - 2)$ (the {\it central critical value}) of the triple product
$L(s,f,g,h)$. We apply this formula, letting $f$ (and thus $\kappa$) vary in a
$p$-adic analytic family ${\bold f}$ of ordinary modular forms (a Hida family).
By modifying Hida's construction of the $p$-adic Rankin-Selberg convolution, we
obtain a generalized $p$-adic measure whose associated analytic function gives
a $p$-adic interpolation of the square roots of the central critical values of
$L(s,f,g,h)$, normalized by certain universal correction factors. The
archimedean correction factor is not determined explicitly.
  This is an example of what appears to be a very general phenomenon of
$p$-adic interpolation of normalized square roots of $L$-functions along the
so-called "anti-cyclotomic hyperplane." We note that the $p$-adic triple
product itself has not been constructed in the half-space $\kappa \geq \lambda
+ \mu$.
