  Let R be a polynomial ring in r variables and D a dual ring upon which R acts
as partial differential operators (classical apolarity). For a type two graded
level Artinian algebras A=R/I, of socle degree j we consider the family of
Artinian Gorenstein [AG] quotients of A having the same socle degree. By
Macaulay duality, A corresponds to a unique 2-dimensional vector space
W(A)=<F,G> of degree-j homogeneous elements (forms) in D, and each such AG
quotient of A corresponds to an element $F_\lambda=F+\lambda G$ in W(A), up to
non-zero constant multiple. Our main result is a lower bound for the Hilbert
function of the generic AG quotient of A in terms of the Hilbert functions of A
and the AG quotients of R determined by F and by G. This result restricts the
possible sequences H that may occur as the Hilbert function for a type two
level algebra A.
  This result may also be viewed as obtaining an optimal lower bound on the
dimensions of the spaces of degree-u partial derivatives of $F+\lambda G$ for a
generic lambda in terms of the corresponding dimensions for the forms F and G.
