  We obtain a combinatorial formula related to the shear transformation for
semi-invariants of binary forms, which implies the classical characterization
of semi-invariants in terms of a differential operator. Then, we present a
combinatorial proof of an identity of Hilbert, which leads to a relation of
Cayley on semi-invariants. This identity plays a crucial role in the original
proof of Sylvester's theorem on semi-invariants in connection with the Gaussian
coefficients. Moreover, we show that the additivity lemma of Pak and Panova
which yields the strict unimodality of the Gaussian coefficients for $n,k \geq
8$ can be deduced from the ring property of semi-invariants.
