The closure conditions, which make a finite set of moment equations
equivalent to the collisionless Boltzmann equation, are investigated for
Gaussian and ellipsoidal velocity distributions working from the com-
plete mathematical expression for the nth-order stellar hydrodynamic
equation, which was explicitly obtained depending on the comoving mo-
ments in a previous paper. First, for a Schwarzschild distribution, it
is proved that the whole set of hydrodynamic equations is reduced to
the equations of orders n = 0,1,2,3, owing to the recurrent form of the
central moments. Furthermore, the equations of order n = 2 and n = 3
become closure conditions for higher even- and odd-order equations, re-
spectively. An arbitrary quadratic function in the peculiar velocities, the
generalised Schwarzschild distribution, is also investigated. Analogous
closure conditions could be obtained from a similar recurrence law for
central moments, but an alternative procedure is preferred, which con-
sists in to expand a generalised ellipsoidal function as a power series of
Schwarzschild distributions with the same mean. Then, due to the linear
nature of the problem, the equivalence between the moment equations
and the system of equations that Chandrasekhar had obtained working
from the collisionless Boltzmann equation is borne out.
