A Rival

I think I never felt jealousy till now. This was not like enduring the endearments of Dr. John and Paulina, against which while I sealed my eyes and my ears, while I withdrew thence my thoughts, my sense of harmony still acknowledged in it a charm. This was an outrage. The love born of beauty was not mine; I had nothing in common with it: I could not dare to meddle with it, but another love, venturing diffidently into life after long acquaintance, furnace-tried by pain, stamped by constancy, consolidated by affection's pure and durable alloy, submitted by intellect to intellect's own tests, and finally wrought up, by his own process, to his own unflawed completeness, this Love that laughed at Passion, his fast frenzies and his hot and hurried extinction, in this Love I had a vested interest; and whatever tended either to its culture or its destruction, I could not view impassibly (Ch. 39).

Lucy witnesses what she calls a "sylvan courtship" between M. Paul and a Justine Marie during a drug-influenced visit to a party.  M. Paul arrives to the party with a young female companion.  Lucy is upset at this sight and leaves the party without speaking to M. Paul.

This is Lucy's first acknowledgement of her love for M. Paul.  Losing Dr. John was not a surprise to Lucy because he valued beauty which she did not have, and she has since seen unattractive character flaws in him.  She is genuinely happy for Polly when Dr. John marries her.  However, she is cannot fathom losing the man with whom she has endured the trials of true affection.  M. Paul's arrival with this unfamiliar face with whom he shares some obvious intimacy has a harrowing effect on Lucy and she leaves without seeking M. Paul for an explanation.  This new interest would be an explanation for the professor's disappearance for several days.  Nevertheless, this companion turns out not to be a love interest but a ward of his.

Lucy also identifies Madame Beck as a rival for M. Paul's attention:

"Dog in the manger!" I said: for I knew she secretly wanted him, and had always wanted him. She called him "insupportable:" she railed at him for a "dévot:" she did not love, but she wanted to marry, that she might bind him to her interest. Deep into some of Madame's secrets I had entered--I know not how: by an intuition or an inspiration which came to me--I know not whence. In the course of living with her too, I had slowly learned, that, unless with an inferior, she must ever be a rival. She was my rival, heart and soul, though secretly, under the smoothest bearing, and utterly unknown to all save her and myself (Ch. 38).

The above passage would explain why Madame Beck encouraged Lucy to spend time with the Brettons.  As long as Lucy was distracted by a love for Dr. John, Lucy cannot pursue M. Paul.  Additionally, the prospect of his marrying a Protestant is unsettling to Madame Beck, who is not in love with M. Paul herself but wants to control her kinsman and take advantage of his selflessness.

Madame Beck made also her private comment, and preferred in her own breast her secret reason for desiring expatriation. The thing she could not obtain, she desired not another to win: rather would she destroy it (Ch. 39).

Nevertheless, M. Paul never expresses an interest in Madame Beck.  At most she is a friend:

"You know not what I have of steady and resolute in me," said he, "but you shall see; the event shall teach you. Modeste," he continued less fiercely, "be gentle, be pitying, be a woman; look at this poor face, and relent. You know I am your friend, and the friend of your friends; in spite of your taunts, you well and deeply know I may be trusted. Of sacrificing myself I made no difficulty but my heart is pained by what I see; it must have and give solace. Leave me!"  (Ch. 41).

He tells her to "be a woman," which is a reason for his lack of interest in Madame Beck; she is not feminine.  Upon first meeting her, Lucy states:  "At that instant she did not wear a woman's aspect, but rather a man's" (Ch. 8).

The above painting is Her First Sorrow by Edward Killingworth Johnson (1825-96).