In BBC Sherlock, Season 4, Episode 3 The Final Problem we learn that:
Sherlock and Mycroft have a sister called Eurus, which means "the god of east wind"
Is this name mentioned in any original story by A. C. Doyle? In what context?
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In the original books, no, a sister is never mentioned. However, in a very roundabout way, the seeds of her origin are contained within the books.
The only sibling mentioned in Doyle's books is Holmes's brother Mycroft, who appears in "The Adventure of the Greek Interpreter", "The Final Problem", and "The Adventure of the Bruce-Partington Plans" and is mentioned in "The Adventure of the Empty House". In the books he appears as much the same character depicted in the TV series - a high up government official who is involved in many aspects of government policy.
The closest mention to 'Eurus' (the sister's name in the TV show) comes in the story "His Last Bow", published in 1917 (set in 1914). The story ends with Holmes' addressing his assistant Doctor Watson on the eve of the First World War:
"There's an east wind coming, Watson."
"I think not, Holmes. It is very warm."
"Good old Watson! You are the one fixed point in a changing age. There's an east wind coming all the same, such a wind as never blew on England yet. It will be cold and bitter, Watson, and a good many of us may wither before its blast. But it's God's own wind none the less, and a cleaner, better, stronger land will lie in the sunshine when the storm has cleared."
In Greek mythology, Eurus was the god of the east wind, one of the four directional Anemoi, or Wind-Gods. In the books, the 'east wind' was used as a metaphor for the destruction that WWI would bring. It seems that in the TV show, it was personified as 'Eurus', another east wind bringing destruction of a similar nature.
The connection is also alluded to in the closing scenes of 'The Lying Detective' (4x02), where John Watson begins to realise that his therapist is not who she seems:
J: Who are you?
E: Isn't it obvious? Haven't you guessed? I'm Eurus.
E: Silly name, isn't it? Greek. Means "the east wind". My parents loved silly names, like Eurus... or Mycroft... or Sherlock.
E: Oh, look at him. Didn't it ever occur to you, not even once, that Sherlock's secret brother might just be Sherlock's secret sister?
Eurus (or Euros) is a completely new creation specifically for the series although she had been 'planned' from the very beginning.
Stephen Moffat to Metro.co.uk
Steven Moffat: ‘We made and, thank God, cut a reference in [series one] episode The Great Game a long time ago where Mycroft is explaining he is smarter than Sherlock, and then Mycroft goes onto say, ‘My sister of course..’ and then gets cut off.
‘It was just a joke, just a passing thing. Thank God we cut that, because we could keep the secret a bit longer. The madness, that we thought would never sustain, of hinting that Sherlock’s got a brother and then pulling, frankly, in the circumstances, the only twist you can – which is it’s actually Sherlock’s sister.’
Co-writer Mark Gatiss to Telegraph.co.uk
But the idea for the character began much earlier, as Gatiss – his co-writer – explains. "It started as a joke years ago there was going to be a line in the first season where we were sketching in the Holmes family," he told Radio Times. "But we thought we won't do too much, and thank God we didn't, because it has given us this place to go."