  We generalise Wesson's procedure, whereby vacuum $(4+1)-$dimensional field
equations give rise to $(3+1)-$dimensional equations with sources, to arbitrary
dimensions. We then employ this generalisation to relate the usual
$(3+1)-$dimensional vacuum field equations to $(2+1)-$dimensional field
equations with sources and derive the analogues of the classes of solutions
obtained by Ponce de Leon. This way of viewing lower dimensional gravity
theories can be of importance in establishing a relationship between such
theories and the usual 4-dimensional general relativity, as well as giving a
way of producing exact solutions in $(2+1)$ dimensions that are naturally
related to the vacuum $(3+1)-$dimensional solutions. An outcome of this
correspondence, regarding the nature of lower dimensional gravity, is that the
intuitions obtained in $(3+1)$ dimensions may not be automatically
transportable to lower dimensions.
  We also extend a number of physically motivated solutions studied by Wesson
and Ponce de Leon to $(D+1)$ dimensions and employ the equivalence between the
$(D+1)$ Kaluza-Klein theories with empty $D-$dimensional Brans-Dicke theories
(with $\omega=0$) to throw some light on the solutions derived by these
authors.
