On a self-ejecting lathe, a mechanism called a lead screw is fixed inside the tailstock quill. The handwheel on the tailstock can be turned clockwise or counterclockwise, thus advancing or retracting the quill. The quill must be advanced far enough to accept an accessory, usually a revolving center, without the center contacting the lead screw. To remove the center, turn the handwheel the opposite way and the quill will reverse into the tailstock, causing the center to contact the lead screw and be pushed out of the quill. In some cases, the Morse taper center may be too long to seat in the quill without contacting the lead screw, and the center will need to be shortened. In other cases, the center may be too short to contact the lead screw and will not eject when the quill is fully retracted. In this case, the accessory center will need to be replaced with one with a longer Morse taper.

A prospective buyer should read the literature supplied with the lathe to determine if the lathe is self-ejecting or not. If no literature is available, shine a light into the tailstock and see if the opening goes all the way through. If the light shines through the tailstock, it is not self-ejecting. Such a lathe has a hollow tailstock, and this can be an advantage. Some operations, such as drilling lamp bases from top to bottom so a cord can be inserted, are easily accomplished with a 3/8" long drill or lamp auger inserted through the end of the tailstock and with the lathe slowly spinning the lamp base. Other operations may also require a hollow tailstock. A lathe with a hollow tailstock will require a knockout rod, which can be inserted into the right end of the tailstock and used to pop the revolving center or other accessory loose from the tailstock quill. Be sure to catch the accessory in your hand. Don’t let it fall on the lathe bed or the floor, because the accessory, such as a revolving center, may be damaged.

From reading your question, it’s obvious you have made a shaving mug of suitable size and proportion. I would suggest that you make another mug, hollow it out, and sand the mug inside and out as much as you can, while leaving the mug attached to the faceplate. A good way to finish the inside of the mug is with a two-part epoxy mix. I suggest using the type which has a working time of twenty minutes and reaches full strength in six to eight hours. There is a quicker setting epoxy, but it never gets as hard as the slower setting material. The epoxy finish should be applied inside the mug in a heavy, even coat. If your lathe is a variable speed and will reduce to a very slow rotation speed, leave the mug on the faceplate, put the faceplate and mug on the spindle, and turn on the lathe at its slowest speed, probably 10 RPM. The rotation will cause the epoxy to be distributed evenly around the inside of the mug. Try to coat the rim of the mug with epoxy also. As soon as the epoxy mixture sets, the lathe can be turned off and the mug allowed to set overnight. From then on, the mug can be sanded and polished inside, the outside finish applied, and the base turned and finished. Be sure to sign your work.