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Zener has suggested a type of interaction between the spins of magnetic ions which he named "double exchange." This occurs indirectly by means of spin coupling to mobile electrons which travel from one ion to the next. We have calculated this interaction for a pair of ions with general spin $S$ and with general transfer integral, $b$, and internal exchange integral $J$.One result is that while the states of large total spin have both the highest and lowest energies, their average energy is the same as for the states of low total spin. This should be applicable in the high-temperature expansion of the susceptibility, and if it is, indicates that the high-temperature Curie-Weiss constant $\ensuremath{\theta}$ should be zero, and $\frac{1}{\ensuremath{\chi}}$ vs $T$ a curved line. This is surprising in view of the fact that the manganites, in which double exchange has been presumed to be the interaction mechanism, obey a fairly good Curie-Weiss law.The results can be approximated rather well by a simple semiclassical model in which the spins of the ion cores are treated classically. This model is capable of rather easy extension to the problem of the whole crystal, but the resulting mathematical problem is not easily solved except in special circumstances, e.g., periodic disturbances (spin waves).