This is Swietoslawski's differential ebulliometer. It is used to determine the temperature difference between a boiling solution of a solvent and a nonvolatile solute and the pure condensing vapor of the solvent. The binary solution is heated in the flask. When it reaches its boiling point it sprays over the lower thermometer, which registers the boiling temperature of the solution. The solution returns to the flask while the vapor passes on to the tube holding the higher thermometer, which registers the boiling temperature of the solvent. The condensed vapor eventually returns to the heating flask so that the concentration remains constant.
The apparatus dates from 1928. The figure and information are from Daniels et al. Experimental Physical Chemistry, 1941.