What does Governor Bellingham mean when he tells Dimmesdale that "the responsibility of this woman's [Hester's] soul lies greatly with you"?

Hester's appeal to Arthur Dimmesdale marks a turning point in the novel. It is probably the first time she has relied on her relationship with the minister for support, and it makes the other men aware that Dimmesdale knows Hester better than they thought. Dimmesdale steps forward with his hand over his heart, again hiding the scarlet letter which he feels upon his breast. This also is related to Chillingworth's comment that he will recognize Pearl's true father by "reading" his heart. Dimmesdale then correctly associates Pearl with the scarlet letter upon her mother's bosom, and he manages to keep the mother and daughter together. Pearl's response is unique at this juncture, taking the minister's hand and placing her cheek against it. This simple gesture is full of meaning, because it implies that Pearl recognizes Dimmesdale as being connected to her. Dimmesdale responds by kissing her on the forehead, in a sense claiming her as his own child. Bellingham, is of course, unaware of the relationship between the three, but the fact that Dimmesdale has advised a different course of action than the one decided upon leaves him responsible.