  We present a new formulation of the equations of motion used in smoothed
particle hydrodynamics (SPH). The spatial resolution in SPH is determined by
the smoothing length, $h$, and it has become common practice for each particle
to be given its own adaptive smoothing length, $h_i$. Consequently, the dynamic
range that may be spatially resolved is greatly increased, but additional
($\nabla h$) terms, which account for the variability of the smoothing lengths,
should appear in the equations of motion in order to satisy conservation
requirements. Previous implementations of SPH have neglected these additional
terms, whereas we have included them. This is achieved by defining a functional
form for the $h_i$s, that depends on inter-particle distances only, and then
deriving the equations of motion using a Hamiltonian formalism. A number of
test calculations are presented. We find that including the $\nabla h$ terms
has no detrimental effect on the ability of SPH to model known problems with
reasonable accuracy. For problems where the energy conservation was rather
poor, including the $\nabla h$ terms results in a dramatic improvement. In
particluar non-conservation of energy during a collision between two polytropes
can occur at the level of 10% when the $\nabla h$ terms are neglected. When the
$\nabla h$ terms are included, then this error reduces to 0.8%.
