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Infrared radial velocities have been used to compute first orbits of the M giants in three southern S-type symbiotic systems. Of the three, Hen 2-173 has the longest orbital period, 911 days, and also has a noncircular orbit with an eccentricity of 0.21. The large value of its mass function suggests that Hen 2-173 may be an eclipsing system. For CL Sco our spectroscopic orbital period of 626 days is essentially identical to the previously determined light variability period of 625 days, and we have adopted the latter. AS 270 has an orbital period of similar length, 671 days, and both CL Sco and AS 270 have circular orbits. Only CL Sco has been extensively investigated previously, and we compare our results with the conclusions of Kenyon & Webbink. We also have examined the period-eccentricity relation for 30 S-type symbiotics. Circular orbits are found for 81% of the systems with orbital periods up to 800 days, while they occur for only 22% with periods greater than 800 days. This distribution is quite unlike that for G and K giants; rather, it is similar to that for barium stars, another type of mass-transfer binary, which also consists of a late-type giant and a white dwarf companion.