  Lam\'e's differential equation is a linear differential equation of the
second order with a periodic coefficient involving the Jacobian elliptic
function ${\rm sn}$ depending on the modulus $k$, and two additional parameters
$h$ and $\nu$. This differential equation appears in several applications, for
example, the motion of coupled particles in a periodic potential. Stability and
existence of periodic solutions of Lam\'e's equations is determined by the
value of its Hill discriminant $D(h,\nu,k)$. The Hill discriminant is compared
to an explicitly known quantity including explicit error bounds. This result is
derived from the observation that Lam\'e's equation with $k=1$ can be solved by
hypergeometric functions because then the elliptic function ${\rm sn}$ reduces
to the hyperbolic tangent function. A connection relation between
hypergeometric functions then allows the approximation of the Hill discriminant
by a simple expression. In particular, one obtains an asymptotic approximation
of $D(h,\nu,k)$ when the modulus $k$ tends to $1$.
