  Considering the cascade decay $D\to c C \to c b B \to c b a A$ in which
$D,C,B,A$ are massive particles and $c,b,a$ are massless particles, we
determine for the first time the shape of the distribution of the invariant
mass of the three massless particles $m_{abc}$ for the sub-set of decays in
which the invariant mass $m_{ab}$ of the last two particles in the chain is
(optionally) constrained to lie inside an arbitrary interval, $m_{ab} \in [
m_{ab}^\text{cut min}, m_{ab}^\text{cut max}]$. An example of an experimentally
important distribution of this kind is the ``$m_{qll}$ threshold'' -- which is
the distribution of the combined invariant mass of the visible standard model
particles radiated from the hypothesised decay of a squark to the lightest
neutralino via successive two body decay,: $\squark \to q \ntlinoTwo \to q l
\slepton \to q l l \ntlinoOne $, in which the experimenter requires
additionally that $m_{ll}$ be greater than ${m_{ll}^{max}}/\sqrt{2}$. The
location of the ``foot'' of this distribution is often used to constrain
sparticle mass scales. The new results presented here permit the location of
this foot to be better understood as the shape of the distribution is derived.
The effects of varying the position of the $m_{ll}$ cut(s) may now be seen more
easily.
