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Caroline has two modes. Mode one we'll call sucking-up-to-Darcy. She's totally hot for him, and he couldn't care less about her, so basically she just hangs on his every word and compliments his every movement. The best is Chapter 10, when she has nowhere to go but to praise how awesome he is at writing—not at putting words together, but at the actual physical act of using a pen.
"How delighted Miss Darcy will be to receive such a letter!"
He made no answer.
"You write uncommonly fast."
"You are mistaken. I write rather slowly."
"How many letters you must have occasion to write in the course of a year! Letters of business, too! How odious I should think them!"
"It is fortunate, then, that they fall to my lot instead of yours."
"Pray tell your sister that I long to see her."
"I have already told her so once, by your desire."
"I am afraid you do not like your pen. Let me mend it for you. I mend pens remarkably well."
"Thank you—but I always mend my own."
"How can you contrive to write so even?"
He was silent. (10.3-14).
Can you read this without cringing? Girl cannot take a hint, even when Darcy drops them like pianos: "there is a meanness in all the arts," he says, "which ladies sometimes condescend to employ for captivation" (8.57). Translation: I'm on to you.
And then there's mode two, which is how she acts to all women (1) who aren't related to her, or (2) who she doesn't want to be related to. She doesn't want her brother to marry Jane partly because Mrs. Bennet is so horrible (our sympathies), and partly because she's constructed this fantasy where she marries Mr. Darcy and her brother marries Miss Darcy. (Keeping it in the family, you know.)
Lizzy is more than capable of shutting her down in person, but she can't do anything about the horrible things Miss Bingley says behind her back, like her "blowsy" hair (8.5), which is nothing to this later Mean-Girls-worthy takedown:
Her face is too thin; her complexion has no brilliancy; and her features are not at all handsome. Her nose wants character—there is nothing marked in its lines. Her teeth are tolerable, but not out of the common way; and as for her eyes, which have sometimes been called so fine, I could never see anything extraordinary in them. They have a sharp, shrewish look, which I do not like at all; and in her air altogether there is a self-sufficiency without fashion, which is intolerable. (46.15)
Criticizing other women to make yourself look good apparently had the same effect in the early nineteenth century as it does today: it just makes you look bad. Sorry, Caro.