Definition of Pathos
In Part 2, Chapter 7, The Underground Man argues that Liza is ruining her life by engaging in sex work, appealing to her emotions through pathos to convince her to give it up. His speech spans pages, showing an outburst of the erratic train of thought that the reader has been privy to the entire time. He insults her sense of self and occupation, saying that she is losing her worth because she is a sex worker:
"What are you giving up here? What are you enslaving? Why, you’re enslaving your soul, something you don’t really own, together with your body! You’re giving away your love to be defiled by any drunkard! Love! After all, that’s all there is!"
He is appealing to her sense of self-worth, disparaging whatever she may think of herself to fit her into the mold of the archetypical "proper woman." He tries to make her feel worthless and undeserving of love in her current state. Of course, this is ironic, as The Underground Man has sought her services and doesn't seem to see any issue in his role here. Additionally, he tries to convince her to be a certain type of woman that society would appreciate more, while he rails against the man that he is supposed to be, according to Petersburg society, for the entire novella.
At the end of his speech, he refers to his tirade as pathos, though sarcastically:
I was so carried away by my own pathos that I began to feel a lump forming in my throat, and . . . I suddenly stopped, rose up in fright, and, leaning over apprehensively, I began to listen carefully as my own heart pounded.
In this moment, the Underground Man pokes fun at his own impassioned speech. He refers to his emotional tirade as pathos, saying that it causes him physical pain. His speech was not a coherent plea to Liza, but an outpouring of his own emotions, and he admits that as he reflects on it two decades later.
Incidentally, the speech did work on Liza. She later comes to The Underground Man's house, admitting that she wants to "get away from ... that place ... once and for all," and turns to The Underground Man for help. He is already shunning society by then, however, and he is already too overtaken by his own paranoia and emotional instability to help her.