  We find an asymptotic expression for the first eigenvalue of the biharmonic
operator on a long thin rectangle. This is done by finding lower and upper
bounds which become increasingly accurate with increasing length. The lower
bound is found by algebraic manipulation of the operator, and the upper bound
is found by minimising the quadratic form for the operator over a test space
consisting of separable functions. These bounds can be used to show that the
negative part of the groundstate is small.
