Looking through Vogel: A Textbook of Practical Organic Chemistry, 3rd ed., p. 115, your mystery glassware most closely resembles a type of manostat, used where an accurate boiling point under reduced pressure is required.
“The device is essentially one for maintaining a constant pressure differential between the pump system and the distillation system [distillation under reduced pressure at a pressure higher than the minimum pressure that the pump will give] ... The liquid in the device may be di-n-butyl phthalate; this permits a pressure range of 1- 15 mm of Hg to be covered, provided of course that the minimum pressure delivered by the pump is less than 1 mm.
MY CAPS
IF HIGHER PRESSURES ARE DESIRED, IT IS MORE CONVENIENT TO USE TWO OR MORE OF THESE REGULATORS IN SERIES THAN TO EMPLOY A LONGER ONE; ALTERNATIVELY, THE MANOSTAT MAY BE CHARGED WITH A SUITABLE VOLUME OF MERCURY, IN WHICH CASE THE BASE SHOULD BE OF THE DRECHSEL BOTTLE TYPE.”
Perhaps the publications of your organic chemist will cast some light on this!
Regards,
MOTOBALL