While on trial in Act 3, Scene 2 for her alleged infidelity, Hermione is given one opportunity to speak in her own defense. In this speech, she acknowledges that Leontes's belief in her infidelity is so entrenched that no amount of reason will change his mind: "Mine integrity, / Being counted falsehood, shall, as I express it, / Be so received." In other words, the use of logos, or the rhetorical appeal to reason, would be useless in this case, since Leontes would only "receive[]" her arguments as falsehoods.
Instead, she uses pathos, or an appeal to emotion, in an attempt to convince Leontes of her innocence, describing her faith in her own honor and asking Leontes to recall her past faithfulness:
I appeal
To your own conscience, sir, before Polixenes
Came to your court, how I was in your grace,
How merited to be so.
Here, instead of appealing to reason, Hermione begs Leontes to remember how she was once "in [his] grace," or in his good graces, and deservedly so. However, Hermione's final, desperate appeal to Leontes's emotions fails to convince him of her innocence. This use of pathos emphasizes the extent of Leontes's delusion. He is so convinced of falsehood that he remains impervious to his past feelings for Hermione.
