The Coke Hat was first developed by Thomas Coke as protection against branches - then usually in a brown colour.   Its protective function necessitated it to be a hard hat.

Ref.:   Money-Coutts, S. A Shot at the Big Time, The Rake, Issue 59 p.115. [145/5006]

The Black Bowler Hat

The black Bowler or Coke is a classic hat that is worn, for instance, in the evening with Black Tie or a Stroller in an urban environment.  It is also part of certain Uniforms worn, e.g. at certain Oxford Colleges and at Royal Ascot, and in the past was part of the outfit of the London bankers.  It also has a place with certain hunts and in equestrian dressage.  This is a stylish and elegant hat.  Like all hard hats is needs careful fitting in the shop.