Soapy is a homeless man who realizes that winter is coming; therefore, he devises a plan to ensure that he will stay warm throughout the winter. He decides that he will get himself arrested so that he can be sent to the prison on Riker's Island and remain in jail throughout the cold months.
So, Soapy breaks a plate glass window of a store and stands waiting. However, the policeman refuses to believe that anyone who remains at the scene would be the perpetrator of such an action. Discouraged, Soapy tries another avenue to his goal of warm. Seeing an attractive young woman on a street five block farther from his first attempt, Soapy adjusts the angle of his hat and plays "the masher." He raising his hat, saying, "Ah there, Bedlia! Don't you want to come and play in my yard?" With the policeman still watching him, Soapy feels confident that the young woman will signal to this policeman. Instead, she finally says,
"Sure, Mike,...if you'll blow me to a pail of suds. I'd have spoke to you sooner, but the cop was watching."
Ironically, Bedelia, whose name is Irish for "exalted one," is a lady of the evening only; that is, a prostitute. Therefore, she avoids the policeman and is happy accompany Soapy. But, he shakes her loose and works again at getting arrested. Of course, in this humorous story, every action has an ironic reaction.
I have no answer for the question just a comment about the previous answer that Bedelia means exhalted.....just interesting i did not see the irony there. Interesting that I have not seen anybody mention the irony in Soapy name. The fact that soap generally has reference to being clean and yet he is a bum.