After section 4.1 giving a brief history of the problem of the cosmological constant, we go up the ladder starting from the standard theory with Λ added (section 4.2), proceeding to the prototype BD model without Λ (section 4.3), and culminating in the prototype BD model with Λ included (section 4.4), where the discussion will concern both the J frame and the E frame. We will face some crucial aspects that Λ has brought into being for the first time. Most remarkable is that the attractor solution in the J frame represents a static universe. This conclusion turns out to be evaded in the E frame, but particle masses are shown to vary with time too much. We then propose in subsection 4.4.3 a remedy in the matter part of the Lagrangian, thus violating the WEP in a manner that, we hope, allows us to remain within the observational constraint. At this cost, however, we are rewarded with a successful implementation of the scenario of a decaying cosmological constant in the E frame, which is now considered to be (approximately) physical. Another point to be noticed is that a physical condition, positivity of the energy density of matter, requires that ∈ = –1, an apparently ghost nature of the scalar field in the J frame, unexpectedly in accordance with what string theory and KK theory suggest. This also entails the condition, and thus is in contradiction with the widely known constraint ω ≳ 3.6 × 103, or ξ ≳ 7.0 × 10-5. A reconciliation with the solar-system experiments will be made only with a nonzero mass of the scalar field.