I assume you are referring to Chapter 3 of the novel, when Hester Prynne and Pearl are being "displayed" as punishment for her sin, and Arthur Dimmesdale is forced to try and persuade her to reveal the name of the father of her child. What is interesting about his appeal is the impact it has on the audience. For we are told that it was so powerful the audience expected Hester Prynne to respond straight away or for the father to be compelled to reveal himself:
So powerful seemed the minister's appeal, that the people could not believe but that Hester Prynne would speak out the guilty name; or else that the guilty one himself, in whatever high or lowly place he stood, would be drawn forth by an inward and inevitable necessity, and compelled to ascend the scaffold.
However, in spite of the power of Dimmesdale's appeal, Hester Prynne is unmoved, merely shaking her head in response to his eloquence. What is important to note, however, is that when she is shouted at by the Reverend Wilson and exhorted to reveal the name, it is Arthur Dimmesdale who she looks at when she responds, and not anyone else:
"Never!" replied Hester Prynne, looking, not at Mr. Wilson, but into the deep and troubled eyes of the younger clergyman. "It is too deeply branded. Ye cannot take it off. And would that I might endure his agony, as well as mine!"
Of course, if you have read the book, you will know that it is actually Dimmesdale who is the father of Pearl, and thus his presence at this scene gives Hester Prynne the opportunity to confirm to him that his secret is safe with her. Note, too, the manner of her response which likewise establishes the tremendous love that she has for him - she wants to suffer alone and would "endure his agony" in addition to her own suffering.