In Book the Third, chapter ten, of A Tale of Two Cities by Charles Dickens, Ernest Defarge has produced a letter before the court at the trial of Charles Darnay. Defarge claims he discovered the letter in Doctor Manette's cell when Defarge led the overthrow of the Bastille.
The letter, written while Manette was in prison, is an account of Manette as a young physician who is kidnapped by a haughty aristocrat named Evremonde and his twin brother who have committed an atrocity against a young girl and her family, especially her brother. Despite his best efforts, Manette was unable to save the girl and assumed he would be free to return to his wife; he was, but only for a very short time. Soon he was whisked mysteriously away to the Bastille, obviously because the Evremondes wanted to ensure Manette's silence.
From his cell, in this letter laboriously written with his own blood, Manette condemns the entire race of Evremondes:
them and their descendants, to the last of their race… I denounce them to Heaven and to earth.
This denunciation breaks Manette's heart to read because he knows his words are condemning his beloved son-in-law, Charles Darnay Evremonde, to a certain death. Even worse, he is also condemning the rest of his little family to the same fate, as Darnay's descendants are included in the curse. What is unclear in chapter ten is why Defarge is so adamant that this letter be read and why he worked so diligently to find it so long ago when he stormed the Bastille.
In chapter twelve of Book the Third, Sidney Carton is in the Defarges' wine shop and overhears the truth that unlocks the mystery: the young woman who died at the hands of the cruel Evremonde brothers was Madame Defarge's older sister, and Madame Defarge has nursed a grudge for decades. Defarge feels badly enough about condemning Charles Darnay to death because Darnay is clearly not the same kind of man as his cruel uncle and his father, but he can understands it must happen. He tries to convince his wife that she should be satisfied by exacting her revenge only on Darnay/Evremonde and spare Lucie and her young daughter, but she will not be dissuaded from her long-held goal:
“Tell the wind and the fire where to stop; not me.”
So, Defarge's motive in betraying Doctor Manette (Defarge's former employer and friend) and putting the dotor's daughter and granddaughter in danger because of their connection to Darney is to help his wife avenge her sister's awful and unnecessary death at the hands of the Evremonde brothers.