  Considering the vorticity formulation of the Euler equations, we partition
the kinetic energy into its contribution from each pair of interacting
vortices. We call this contribution the "interaction energy". We show that each
contribution satisfies a reciprocity relation on triples of vortices:
$\boldsymbol{A}$'s action on $\boldsymbol{B}$ changes the interaction energy
between $\boldsymbol{B}$ and $\boldsymbol{C}$ in an equal and opposite way to
the effect of $\boldsymbol{C}$'s action on $\boldsymbol{B}$ on the interaction
energy between $\boldsymbol{A}$ and $\boldsymbol{B}$. This result is a
curiously detailed accounting of energy flow, as contrasted to standard
pointwise conservation laws in fluid dynamics. This result holds for all
triples of points $\boldsymbol{A},\boldsymbol{B},\boldsymbol{C}$ in two
dimensions; and in 3 dimensions for all points $\boldsymbol{A},\boldsymbol{C}$,
and all closed vorticity streamlines $\boldsymbol{B}$. We show this result in 3
dimensions as a consequence of an interaction energy flow around
$\boldsymbol{B}$ that is a function only of the triple
$(\boldsymbol{A},\boldsymbol{b}\in \boldsymbol{B},\boldsymbol{C})$, a result
which may be of independent interest.
