I first saw the picture titled "Madame X", at the Met in NY in 1997.
There's something about it, that I find fascinating.
It's a tale of society and morals. Originally Sargent painted the shoulder strap on the left, falling down her arm. Society was so scandalized by this, that he redid the photo, changing the position of the strap. Which explains the difference in the detailing between them. Today we wouldn't even notice it. He also changed the title to "Madame X" to try to hide the name of the subject, although of course everyone recognized her.
Later in her life, additional portraits of Gautreau were painted.
Which do you prefer?