Let f ^ M be a differentiable family of compact complex manifolds Vi=nrH.t) on M={teRm | |/| Vt, t e M, of a differentiable family i% —> ~f —s» M was studied. The notion of transportable forms was introduced, and those of extendible and co-extendible forms (see Definition 2.1 below). It was shown in (4) that dim H?s(Bt) is locally constant at a point t0 e M if and only if all harmonic forms y e Hr's(Bto) are transportable. This was shown to be equivalent with the hypothesis that all harmonic forms of the given type at t = t0 are extend- ible and co-extendible (Theorem 2.1 below). The purpose of this paper is to deduce conditions for all harmonic forms of a given bidegree at t = t0 to be extendible in the given family. This will automatically give conditions for co-extendibility, and thus for the constancy of dim H?is(Bt) in a neighborhood of t0 e M. Applications and a further analysis of the conditions (which can be considered as a statement of the vanishing of certain obstructions) are planned to be given in a subsequent paper. In §2 some notations are introduced and the basic result of (4) is recalled. For further details see (4) and, for a general background, (5) and (6). The trivial exten- sion of the harmonic theory of forms with values in the holomorphic tangent bundle T'0 of VQ to one for forms with values in ^ = F? © To is discussed in §3. The structural forms <pt and it which determine (each) the complex structure on Vt, are discussed in §4. §5 gives a development of the form (it, and in §6 the differential equations satisfied by an extension (resp. a co-extension) of a harmonic form are considered. The differential for an extension is seen to be equivalent with an equation with an initial value which is a closed form, and an integrability condition. A solution for the integral in the form of a con- verging series is obtained in §7, and the integrability condition is discussed in §8. Sufficient conditions for dim H%'s(Bt) to be locally constant are derived in §9, and