This is not a kerosene lamp in the usual sense, but rather a vaporizer. It was patented by the Vapo-Cresoline Company of New York in the mid 1880s after the company was started in 1879. Kerosene was used to heat a burner filled with a substance that was then used as a vaporizer to cure things like whooping cough and other respiratory ailments. This Cresoline substance was made with carbolic acid.
Basically, this was quack medicine. Vapo-Cresoline did not cure anything, and in fact put people - especially children - in danger, as their conditions would continue to deteriorate while they believed they were being cured. And yet somehow the company expanded and stayed in business up until the early 1950s, by which time the burners were electrified.
These Vapo-Cresoline "vaporizers" came as kits in boxes, and kits these days are almost always missing pieces, as yours is. An intact burner set might bring $35-$40, while most are worthless. Unfortunately, they're pretty (like most things form the 1880s and 1890s) and therefore attractive and curious enough to make a lamp collector part with a few bucks.
