I think that part of a letter that Jimmy would write would focus on why he had to arrest Bob. The note that Jimmy writes to Bob when he is arrested would be a starting point as to why Jimmy would be disturbed: "Bob: I was at the appointed place on time. When you struck the match to light your cigar I saw it was the face of the man wanted in Chicago. Somehow I couldn't do it myself, so I went around and got a plain clothes man to do the job. JIMMY." For Jimmy, his letter to Bob would center about how he felt agonized in arresting him. In the line, "I couldn't do it myself," there might be the basis of a letter to Bob.
Jimmy would talk about how their friendship as kids, the personalities they both had, experienced a change as they became older. When Bob talks about how Jimmy was a bit of a "plodder," it might belie the fact that Bob has embraced breaking the law in order to find success. The letter that Jimmy writes to Bob might delve into this reality, one in which Jimmy has recognized the need to accept the law no matter what the cost. Jimmy's allegiance to the loyalty supersedes his loyalty to Bob, something that is evident in both the note he wrote to him as he was arrested, as well as the agony he feels now that he had to make the arrest. The letter to Bob would talk about how Jimmy felt agonized at doing what he did. Perhaps, it might involve a level of pleading to Bob that he would understand, or it might be more informative in terms of explaining his actions, but I think that the letter to Bob would be rooted in much of what the note to Bob indicated.
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