  We study the systolic area (defined as the ratio of the area over the square
of the systole) of the 2-sphere endowed with a smooth riemannian metric as a
function of this metric. This function, bounded from below by a positive
constant over the space of metrics, have the standard metric $g\_0$ for critic
point, although this one do not achieve the conjectured global minimum : we
show that for each tangent direction to the space of metrics at $g\_0$, there
exists a variation by metrics corresponding to this direction along which the
systolic area can only increase
