On page 30,Higgins says that Eliza will be hit with a broomstick if she doesn't cooperate.Seconds later,he offers her a handkerchief.
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Higgins is passive aggressive throughout the play. He inflicts pain and discomfort on Eliza, only to offer a semblance of "sympathy" later on. I think his offering of the handkerchief is to alleviate his own guilt rather than help Eliza. Higgins hence shows a condescending impetuousness to people that are weaker than him in social standing.