In this paper, we continue the study of which $h$-vectors $\H=(1,3,...,
h_{d-1}, h_d, h_{d+1})$ can be the Hilbert function of a level algebra by
investigating Artinian level algebras of codimension 3 with the condition
$\beta_{2,d+2}(I^{\rm lex})=\beta_{1,d+1}(I^{\rm lex})$, where $I^{\rm lex}$ is
the lex-segment ideal associated with an ideal $I$. Our approach is to adopt an
homological method called {\it Cancellation Principle}: the minimal free
resolution of $I$ is obtained from that of $I^{\rm lex}$ by canceling some
adjacent terms of the same shift.