Once the piece is dried, it is ready to be re-mounted on the lathe so that turning can be completed.  The preferred method is to mount the bowl by its bottom portion, beginning work on the interior of the bowl.  Once the inside has been shaped, the outside can be shaped to follow the interior of the form.  Try for yourself and see what works best for you. 
Additional Web Resources:

• Turning a Dry Elliptical Bowl Round (DIY Network)

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Jump to another section in this tutorial:

• Green Wood Bowl Turning, Part 1 - Introduction
• Green Wood Bowl Turning, Part 2 - Preparation
• Green Wood Bowl Turning, Part 3 - Drying Methods
• Green Wood Bowl Turning, Part 4 - Monitoring the Drying Process
• Green Wood Bowl Turning, Part 5 - Finishing