  There are many methods used to compute the classical gravitational field in
astrophysical simulation codes. With the exception of the typically impractical
method of direct computation, none ensure conservation of angular momentum to
machine precision. Under uniform time-stepping, the Cartesian fast multipole
method of Dehnen (also known as the very fast tree code) conserves linear
momentum to machine precision. We show it is possible to modify this method in
a way that conserves both angular and linear momenta.
