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A holomorphic foliation on $\mathbb P^2_{\mathbb C}$, or a real analyticfoliation on $\mathbb{P}^{2}_{\mathbb{R}},$ is said to be convex if its leavesother than straight lines have no inflection points. The classification of theconvex foliations of degree $2$ on $\mathbb P^2_{\mathbb C}$ has beenestablished in $2015$ by C.~\textsc{Favre} and J.~\textsc{Pereira}. The mainargument of this classification was a result obtained in~$2004$by~D.~\textsc{Schlomiuk} and N.~\textsc{Vulpe} concerning the real polynomialvector fields of degree $2$ whose associated foliation on$\mathbb{P}^{2}_{\mathbb{R}}$ is convex. We present here a new proof of thisclassification, that is simpler, does not use this result and does not leavethe holomorphic framework. It is based on the properties of certain models ofconvex foliations of $\mathbb P^2_{\mathbb C}$ of arbitrary degree and of thediscriminant of the dual web of a foliation of $\mathbb P^2_{\mathbb C}$.

Une nouvelle d\'emonstration de la classification des feuilletages convexes de degr\'{e} deux sur $\mathbb P^2_{\mathbb C}$