Trying to understand the original use for the saucer shaped large linhof ball-head.
It is of the largest size as you can see in the photo for scale.

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It has quite an odd function however as the handle/shaft sticking from the base controls the tension on the ball-head! This means on a normal tripod with a center column you could not use it.
The base is the same dimensions as the cap for the Heavy-Duty tripod(and other large camera mounts), so it can be mounted there and the center column removed and it would work.

This makes me wonder if it had some specialty application or maybe was housed on an articulating arm on one of the big studio stands...