The variational principle of Bohm and Pines for the ground-state energy of the electron gas with uniform neutralizing positive-charge background, employing auxiliary variables, is reviewed as an illustration of the past use of auxiliary variables and as an example of the type of physical system to which the variational principle of this paper can be applied. We then develop this variational principle for the logarithm of the partition function of a physical system at nonzero temperatures, employing auxiliary variables. The variational principle contains a trial Hamiltonian H′ in an extended Hilbert space. For H′ equal to its optimal value H, the variational expression ln Q′ is equal to the logarithm of the partition function ln Q. For H′ ≠ H, it is shown that ln Q ≥ ln Q′ for H′ – H sufficiently small or temperatures sufficiently high, or for sufficiently low temperatures when an additional assumption is made, which reduces to one made by Bohm and Pines when applied to the electron gas. The variational expression ln Q′ contains more complicated trace formulas than are usually encountered in quantum statistical mechanics; one possible method of evaluation is sketched leading to a simpler approximate formula for ln Q′. Corrections to the variational approximation for ln Q are provided by the second- and higher-order terms in a certain perturbation expansion of ln Q. The variational principle developed here implies the variational principle of Bohm and Pines in the zero-temperature limit; in the ``no auxiliary variable'' limit it reduces to a modified form of Peierls' variational theorem. It is shown how the variational principle can be applied to any physical system containing charged particles in which the long-range collective effects of the Coulomb interaction are important.