"Just missed you boys once, over there in the Khartoum business," the genial skyfarer informed them, "trying to make it out of town a couple steps ahead of the Mahdi's army—saw you sailing overhead, wished I could've been on board, had to settle for jumpin in the river and waiting till the clambake subsided a little."
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"the Khartoum business"
This would be the Siege of Khartoum from March 1884 to January 1885. The moment Ray is talking about must have been in January 1885 seeing as how he was trying to "make it out of town a couple steps ahead of the Mahdi's army." Which means the Chums of Chance were flying around the world eight years previously. Darby certainly wouldn't have been a member of the crew then unless they allow toddlers on board. Or unless, of course, the Chums are ghosts or angels! I'm still hoping for that to be true.
What possible reason could the Chums of Chance have for involving themselves in Egyptian and Sudanese business? They could be mercenaries. They could be operatives for the interests of the United States. They could be angels ala Michael Landon in Highway to Heaven just going around helping make the world a better place. Or maybe they're more like Sam in Quantum Leap, flying about re-adjusting history like some omniscient autocorrect.
We haven't learned much about Ray Ipsow yet so I'm not particularly interested in why he was in Khartoum. Unless it has something to do with rival aeronaut clubs trying to spread their influence across the globe in some kind of weird Earth-civilization manipulating game! Maybe they're playing real life versions of Diplomacy or Risk!
"the Khartoum business"
I know I just explained this but only in the perspective of what it meant to the Chums of Chance and Ray Ipsow. The real reason Pynchon uses it as a location is because it's another example of British imperialism across the world, especially of British Imperialist failures like all the best plays performed at the Chicago World's Fair.
The Khartoum business and British imperialism across the globe is complicated by the fact that Britain was working on the behalf of Egypt, which wasn't a colony but a protectorate, to retreat from Sudan. I don't know how a protectorate really differs from a colony but I suspect it amounts to nearly the same thing. I suppose the only difference is that Egypt's interests paralleled British interests enough that Britain didn't have to set up a British government. But they still sent British administrators and leaders to accomplish British goals while working with the Egyptians. It's all too complicated for somebody who never bothered with history because he was reading every book written by Steinbeck, Kesey, Heller, and Vonnegut!
Egypt was in control of Sudan when the Mahdi's army revolted. The British were all, "My, my, old chap Egypt. Perhaps you should just move along. Wouldn't want to get wrapped up in a war for that miserable place, wot?" And the Egyptians were all, "Well, I guess since we're your protectorate (whatever that means), we'll follow your advice and evacuate." And the British were all, "We'll send you our best man for the job, meaning a man we'd really like to separate ourselves from a bit so he's off to Khartoum to help you boys out! Cheerio!" Also the man they sent, General Charles George Gordon, had previously been Governor-General of Sudan. So see? More imperialism! No wonder the Sudanese decided the Mahdists were the team to follow. Who wants the British to set up control again? Or their lackeys the Egyptians?! Better to go with this new team, these Islamist guys!
Gordon thought the Mahdists needed to be defeated so he sort of drug his feet on the evacuation thing and began preparing for battle. But Britain and Egypt were staunchly against supporting his fight and kept trying to remind him to evacuate the city. Eventually it was too late and the city was lying in siege by the Mahdi's army.
Remember the Siege of Paris? I bet that's why all these balloonists and aeronauts were here. They were delivering messages and trying to provide aid for the people trapped within the city limits.
Eventually, the British agreed to send help. But upon hearing of reinforcements on the way, the Mahdi's army pressed the siege and invaded the city. They killed every last soldier, many civilians, and, of course, Charles George Gordon. The success of the invaders rebelling against the British and their Egyptian protectorate, even if the Mahdi's regime was one of oppressive Sharia law, is almost certainly why Pynchon mentions this event. Like the plays at the Chicago World's Fair, this was a moment in history where the people oppressed by British colonialism rose up and scored a win.
P.S.
I do not claim to be a historian. If you want to know the intricacies of any historical event I reference and write about, please research it on your own terms! I'm lazy and tend to say things in my own words so much so that I'm probably completely bungling the explanation of events! I just want to explain enough so that one (that one being me, really) can get a sense of why Pynchon mentions any of the things he mentions. He doesn't just use mentions of historical events as filler or color. There's a reason for everything. He's building a sense of the world in 1893 in a way that allows the reader to absorb as much or as little as they choose.