  In this and the accompanying paper the problem of the maximally achievable
elongation in a tokamak is investigated. The work represents an extension of
many earlier studies, which were often focused on determining the elongation
limits due to (1) natural elongation in a simple applied pure vertical field or
(2) axisymmetric stability in the presence of a perfectly conducting wall. The
extension investigated here includes the effect of the vertical stability
feedback system which actually sets the maximum practical elongation limit in a
real experiment. A basic resistive wall stability parameter (gammatau) is
introduced to model the feedback system which although simple in appearance
actually captures the essence of the feedback system. Elongation limits in the
presence of feedback are then determined by calculating the maximum elongation
against n=0 resistive wall modes for fixed gammatau. The results are obtained
by means of a general formulation culminating in a variational principle which
is particularly amenable to numerical analysis. The principle is valid for
arbitrary profiles but simplifies significantly for the Solov'ev profiles,
effectively reducing the 2-D stability problem into a 1-D problem. The
accompanying paper provides the numerical results and leads to a sharp answer
of how much elongation is too much?
