Ganya starts laying it all out there. And we mean all—turns out dude's a regular Sherlock Holmes. Or maybe Jerry Springer. Either way, he's totally figured this whole paternity situation out.
So, basically, the deal is that there is no possible way that Pavlichev is Burdovsky illegitimate dad—he was traveling abroad for a year and a half before the birth, and Mrs. Burdovsky's never left the country.
Actually, the reason that Pavlichev was so nice to that family is that Mrs. Burdovsky had a sister who was Pavlichev fiancée—but she died young, so he kind of kept up the connection to the family through the sister. Also Pavlichev had a real soft spot for sick kids, so when Burdovsky was little, Pavlichev grew close to him because of his stutter and limp. This is also why, later, he semi-adopted Myshkin. So as he was growing up, Burdovsky and those around him got the wrong sense that he might be Pavlichev's son. But yeah, there's no there there on the illegitimate kid front.
Burdovsky hears all this and just loses it. He even gives back the small amount the prince had already sent him and then storms off.
Everyone else is really grossed out to be watching this sleazy scene play out. It's really got everything the 19th century isn't too into: sex, extra-marital sex, and open talk about money. Just ewww all around for these guys.
Myshkin reacts in a bizarre way to Ganya's revelations—he feels terrible that he accused Burdovsky and his crew of being corrupt liars and starts apologizing left and right. He offers money and friendship.
Ippolit is all, yeah, that's a nice offer, since no honest person could accept it from you now. Which, wow, for nihilists, they sure do have a very complicated sense of honor. We're thinking this is meant as a subtle mockery of the whole we-don't-believe-in-morality stance.
Meanwhile, all this time, Mrs. Epanchin has been sitting and stewing, and now suddenly she just explodes and tells off the whole party.
Everyone gets yelled at. The nihilists because they are stupid kids who say they don't believe in gratitude, but expect Myshkin to give them money because he feels gratitude to Pavlichev, and in general because their belief system could never work in the real world where people have to live together and have to have some rules to conduct themselves by. Myshkin gets yelled at for apologizing to these lowlifes and for not having the cojones to kick them to the curb.
The rest of the party is kind of freaking out watching Mrs. Epanchin lose it because she seems way out of control.
Finally, she turns to leave, and in a final moment of rage grabs Ippolit's arm.
He is basically on death's door, and collapses in a coughing fit (he's got consumption, which is old-timey speak for tuberculosis).
Immediately, Mrs. Epanchin feels bad for him—and again, check out how quickly she responds to a suffering human being. The novel has a lot of these moments, basically trying to point out that there is a natural morality that's embedded in humans. We pity and empathize with pain in others, for example, however mad we might be.
Ippolit declares that he has about two weeks to live, so he might as well say a few last words while in good company. Which, yeah, might as well, since back then t.b. was almost always fatal.
Everyone gathers around, and Myshkin orders some tea for them all.