Your hub is cracking and needs replacement.
Kinda hard with the photo that close. Is that hub on, or from a Schwinn American? It looks like an Excel Hub. Kind of a cheap cheesy hub that Schwinn and other USA bike companies used during the 60's.
Schwinn had this idea that American's would like to buy "Something actually made in America". Along came The American model. Unfortunately, many of the bicycle parts supplies were dropping away from making any bicycle parts. Diamond Chain was gone from the bicycle industry, you had New Departure drop away in the late 1950's. Schwinn was forced (by pricing and availability) into using the Excel front hubs. It all came to an end when the Federal Trade Commission contacted Schwinn and told them they were in conflict with the Federal Truth in Advertising Laws since not every single part on the American bike was actually built in America. Never mind that not one chain company located in the U.S. was still making bicycle chains. Schwinn used the Union brand German made chain on the American models. The American went away as a result.
John