You might want to check it but the adapter on my lathe has a sweet spot, depending on where it's oriented. You can have .005" of runout in 1 place and as little as .0001' in the sweet spot.
Lathes of your size have morse taper 5 tapers on the spindle. As seasicksteve said, its for a center in your spindle, or to use an mt3 drill or tool in a morse taper 5 machine. Had a lot of them at work for drilling "small" 3/4" holes on a absolutely massive radial arm drill
Generally, the center for the headstock is left soft, so it can easily be recut to run true, these centers usually have a shallow ring cut near the 60 degree point so that they may be distingushed from the hard tailstock center.
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