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Any $r$-dimensional subbundle of the cotangent bundle on an $n$-dimensional manifold $M$ partitions $M$ into subsets $M_0,...,M_m$ ($m$ being the minimum of $r$ and $C(n-r,2)$, the combinations of $n-r$ things taken 2 at a time). $M_i$ is the set on which the first derived systems of the subbundle has codimension $i$. In this paper we prove the following: Theorem. Let $s\ge 2$ and let $Q$ be a generic $C^s$ $r$-dimensional subbundle of the cotangent bundle of an $n$-dimensional manifold $M$. The codimension of $M_i$ is given...