Character Analysis
Bilham, or "Little Bilham" as he is oh-so-adorably called throughout the book, is a starving artist who is close friends with Chad Newsome. Strether first meets Bilham while Bilham is housesitting at Chad's apartment. Bilham gives Strether's his first exposure to Chad's interesting circle of Paris friends, and it's not long after meeting Bilham that Strether begins to worry that life has passed him (Strether) by.
On some level, Bilham seems to be a younger version of Strether himself, and he can see this especially when Bilham talks about his experience in Paris: "They've simply—the cannibals!—eaten me, converted me if you like, but convereted me into food. I'm but the bleached bones of a Christian" (5.1.54). This statement could have come word-for-word out of Strether's mouth, since it's Strether who also feels like Paris has totally consumed him and that he'll never want to live anywhere else again. Sure, it's more poetic than Strether could muster, but there's a similar kinda boniness about the guy.
Like many of Chad's friends, though, Bilham is also dodgy when it comes to answering Strether's questions about Chaddy-boy's lifestyle. When asked to describe Chad's relationship with Madame de Vionnet—in other words, whether they're sleeping together—Bilham replies, "It's, upon my word, the very finest thing I ever saw in my life, and the most distinguished" (6.3.70). Strether takes this answer to mean that Chad and Madame have a platonic relationship, but Bilham means something very different. It's unclear whether he's intentionally misleading Strether, but the odds are that he is.
Although Strether admires Bilham's artistic lifestyle, Bilham is also a shy little man, much like Strether himself. He's afraid to approach a girl as high-class as Jeanne de Vionnet, for example, because he feels like he's not good enough for her. Or to put it differently, he says, "Certainly I'm good enough. We're good enough, I consider, nous autres, for anything. But she's too good" (6.3.46). Aw, single tear.
Strether, though, can't bear to watch Bilham pass up opportunities because of fear. He sees himself in Bilham and wants (nay, needs) this kid to live life to its fullest, especially while he's still young. He tries to set him up with Mamie Pocock at the end, and we can only hope Lil' Bill does a better job of it than our Strether.
Bilham's Timeline