Final answer:
In Act III of The Crucible, Judge Hathorne and Danforth are most troubled about the possibility that the court's authority may be undermined.
Step-by-step explanation:
In Act III of The Crucible, Judge Hathorne and Danforth seem to be most troubled about the possibility that: c) the court's authority may be undermined.
In this act, there are increasing doubts about the validity of the witch trials. Several influential individuals, including the wife of Governor Phipps, are accused, and the educated elite of Boston starts pressuring the Governor to consider setting aside evidence based on spectral testimonies.
Hathorne and Danforth are concerned that if these doubts and suspicions about the court's authority continue to grow, it may undermine their power and legitimacy.