  In order to calculate the multiplicity of an isolated rational curve C in a
local complete intersection variety X, i.e. the length of the Hilbert scheme of
X at [C], it is important to study infinitesimal neighborhoods of the curve in
X. This is equivalent to infinitesimal extensions of P^1 by locally free
sheaves. In this paper we study infinitesimal extensions of P^1, determine
their structure and obtain upper and lower bounds for the length of the local
rings of their Hilbert schemes at [P^].
