This part of the novel is found in chapter 5. The purpose of Sybil Vane's character in the plot of The Picture of Dorian Gray is to represent that unspoiled innocence that Wilde often uses as a motif in a lot of his works. As such, that very innocent will inevitably be shattered by the influence of Dorian in the same manner that Dorian's own innocence was breached by the influence of Harry Wotton.
A big part of Sybil's innocence is seen in the outlandish fantasies that she proposes to herself and others, fantasies that she takes for granted and in which she seems to excel.
When the time came for James, Sybil's brawny and tough brother, to go to Australia to try his luck at the gold diggings, the young woman viewed this sad reality as the episode of an epic tale in which her brother not only will find the love of his life, but will also become rich, be victorious against the odds and live happily ever after.
For he was not to remain a sailor, or a super-cargo, or whatever he was going to be. Oh, no! [...] He was to leave the vessel at Melbourne, [...]Before a week was over he was to come across a large nugget of pure gold [...]
Then Sybil changes her mind. Her brother is too good for the gold mines. These are horrid places, for drunks and runaways, she claims.
He was to be a nice sheep-farmer [...], he was to see the beautiful heiress being carried off by a robber on a black horse, and give chase, and rescue her. Of course, she would fall in love with him [...] they would get married, and come home, and live in an immense house in London.
All of these divergent and yet rich fantasies denote a woman who may not be entirely in her senses. Certainly, there is something a bit off about Sybil's dreams. She doesn't consider the reality that is right in front of her. It is almost as if she sends out a fantasy into the universe for some miracle to make them happen.
She is oblivious to Jim's rough nature; he would never be able to be a sheep farmer, nor a commonplace man about London. He is far from a rescuer of a damsel in distress. He may never become rich, and his status as the out-of wedlock son of a former actress and brother to a current one places him at the bottom of the social totem pole. For Sybil to omit all of these facts in favor of a ludicrous fantasy tells a lot about someone. It is no surprise that once she had her first taste of reality--when Dorian leaves her--she could not take it, and thus she commits suicide as a result.