A set of differential equations is established that determine the posi
tions of the tielines for phase separation at constant temperature, pr
ovided the equation of state for a binary mixture is given in analytic
al form. Despite the fact that a large amount of work has been done on
this problem, there is little information on the general theory, exce
pt for some papers that were written at the turn of the century. In mo
st cases tielines ere calculated by a ''brute force'' method, i.e. aft
er the problem is formulated a non-linear simultaneous equation-solvin
g program is called upon, with little or no attention to the specifics
of the problem on hand. It is shown that there exists a very useful r
elation between the orientation of the tieline and the tangents to the
binodals at each end. This relation persists near the almost pure liq
uid limit despite the fact that the Helmholtz free energy is a singula
r function in this case. In addition, a differential equation is given
for the binodal as well as a theorem on the termination of a set of t
ielines in the neighborhood of a three-phase (or triple) point. The eq
uations given here are different from the Gibbs Konowalow expressions.
The latter hold at constant concentration, while the equations presen
ted here are at constant temperature.