  For a discrete function $f\left( x\right) $ on a discrete set, the finite
difference can be either forward and backward. However, we observe that if $
f\left( x\right) $ is a sum of two functions $f\left( x\right) =f_{1}\left(
x\right) +f_{2}\left( x\right) $ defined on the discrete set, the first order
difference of $\Delta f\left( x\right) $ is equivocal for we may have $ \Delta
^{f}f_{1}\left( x\right) +\Delta ^{b}f_{2}\left( x\right) $ where $ \Delta
^{f}$ and $\Delta ^{b}$ denotes the forward and backward difference
respectively. Thus, the first order variation equation for this function $
f\left( x\right) $ gives many solutions which include both true and false one.
A proper formalism of the discrete calculus of variations is proposed to single
out the true one by examination of the second order variations, and is capable
of yielding the exact form of the distributions for Boltzmann, Bose and Fermi
system without requiring the numbers of particle to be infinitely large. The
advantage and peculiarity of our formalism are explicitly illustrated by the
derivation of the Bose distribution.
