where we have introduced the isothermal sound velocity v$_{T}$ . The sum rules in (7) and (11) are general and may be viewed as two constraints which S ( q , ω ) must always satisfy. The f − sum rule is used frequently in discussions of inelastic neutron scattering as a check on the experimental data. As we discuss in section III, the compressibility sum rule has special significance when we discuss the two-fluid hydrodynamic region, as first emphasized by Nozi` eres and Pines [23]. The Landau two-fluid equations for an isotropic (such that χ$_{nn}$ ( q , ω ) = χ$_{nn}$ ( q, ω )) super- fluid (Bose or Fermi) describe the dynamics when collisions are strong enough to produce local equilibrium. Using the non-dissipative two-fluid equations, one finds the density re- sponse function (see Refs. [11, 12] and page 138 of [3]), where we have defined a new velocity Here, ¯ s$_{0}$ ≡ S$_{0}$/Nm is the equilibrium entropy S$_{0}$ per unit mass and ¯ c$_{v}$ ≡ T ( ∂ ¯ s/∂T )$_{ρ}$ is the specific heat per unit mass at constant volume. The equilibrium superfluid and normal fluid densities are denoted by ρ$_{s}$$_{0}$ and ρ$_{n}$$_{0}$ , respectively. In (12), u$_{1}$ and u$_{2}$ are the well- known exact first and second sound velocities given by Landau’s hydrodynamics. For later purposes, we note that these velocities satisfy the exact relations where v$_{T}$ is defined in (11) and the adiabatic sound velocity is defined as We note that thermodynamic relations [24] show that, quite generally, the ratio of the specific heats can be related to the isothermal and adiabatic sound velocities, 8 γ ≡ ¯ c$_{p}$ ¯ c$_{v}$ = v 2 ¯ s v 2 T . (16) v 2 ¯ s ≡ ∂P ∂ρ ∣ ¯ s . (15) u 2 1 + u 2 = v 2 + v 2 ¯ $_{s}$, u 2 $_{1}$u 2 = v 2 $_{T}$v 2 = v 2 ¯ s v 2 γ , (14) v 2 ≡ T ¯ s 2 0 ¯ c$_{v}$ ρ$_{s}$$_{0}$ ρ$_{n}$$_{0}$ . (13) χ$_{nn}$ ( q, ω ) = nq 2 m ω 2 − v $^{2}$q2 ( ω 2 − u 2 $_{1}$q$^{2}$)( ω 2 − u 2 $_{2}$q$^{2}$) , (12) III. DYNAMIC STRUCTURE FACTOR IN THE TWO-FLUID REGIME