Let us remember that Aunt Harriet was the relative of Mrs. Ansley that was famous for sending her sister out to the Forum deliberately after sunset on the pretext of getting her to gather a "nighblooming flower for her album." However, the real reason was that they were both in love with the same man, and Aunt Harriet acted out of jealousy to ensure that her sister caught the dreaded Roman fever and died as a result.
The importance of this story is that it acted as the inspiration for what Mrs. Slade did to her childhood friend. Note how greatly this story of Aunt Harriet impressed her:
Well, I suppose that was why the story of your wicked aunt made such an impression on me. And I thought: "There's no more Roman fever, but the Forum is deathly cold after sunset--especially after a hot day. And the Colosseum's even colder and damper."
So it is that Aunt Harriet's story acts as the inspiration for Mrs. Slade to write a letter from her fiance to Mrs. Ansley organising a secret tryst in the Colosseum to try and kill her off out of jealousy. Fortunately, however, from what we are told, we are given no reason to assume that this pattern of jealousy is repeated down the generations to the daughters of Mrs. Slade and Mrs. Ansley. From the little we are told of them, we have no reason to suspect that Jenny is planning to try and kill Barbara because of her love for the same aviator that it looks as if Barbara is going to marry.