  We define the notion of {\it Dwork family of logarithmic $F$-crystals}, a
typical example of which is the family of Gauss hypergeometricdifferential
systems, viewed as parametrized by their exponents of algebraic monodromy. The
$p$-adic analytic dependence of the Frobenius operation upon those exponents,
is Dwork's "Boyarsky Principle". We discuss, in favorable cases, the $p$-adic
analytic continuation of the unit root $F$-subcrystal in the open tube of a
singularity, uniformly w.r.t. the exponents. We obtain a conceptual proof of
the Koblitz-Diamond formula $p$-adically analog to Gauss' evaluation of
$F(a,b,c;1)$.
