Why does Danforth rejoice when Proctor confesses, and what does this imply about the purpose of the trials and Danforth himself?

 
If Proctor confesses to witchcraft, his confession will seem to legitimize both the executions that have already taken place as well as those scheduled to take place on this day. This is why Danforth rejoices when Proctor confesses; it seems to confirm everything else the court has done, no matter how unpopular. Reverend Parris points out that those who will hang today are not like the others who have previously been hanged. He says,

Rebecca Nurse is no Bridget that lived three year with Bishop before she married him. John Proctor is not Isaac Ward that drank his family to ruin. I would to God it were not so . . . but these people have great weight yet in the town.  

The townspeople are beginning to lose faith in the court, and officials fear they may riot if the hangings occur. Parris requests a postponement of the executions, as does Hale.  However, Danforth refuses all such requests, saying,

Twelve are already executed; the names of these seven are given out, and the village expects to see them die this morning. Postponement now speaks a floundering on my part; reprieve or pardon must cast doubt upon the guilt of them that died till now. While I speak God's law, I will not crack its voice with whimpering. If retaliation is your fear, know this—I should hang ten thousand that dared to rise against the law, and an ocean of salt tears could not melt the resolution of the statutes.

In other words, according to Danforth, because twelve people have already been hanged for this crime, they must continue with the hanging of these seven. If they do not, it will look like they are suggesting that these condemned might really be innocent, which would lead to questions about the innocence of the others who have already been executed. This will make the court look not only ridiculous but also murderous. It will also completely undermine Danforth's own authority, something he is absolutely unwilling to risk. In fact, he would rather—in his own words—hang ten thousand people simply because they dared to protest the court's actions. He is resolved to hang Proctor because it is the only way to maintain his authority and power.