How did Batman and the Joker become arch-enemies?

gamesradar.com:

The Joker is arguably Batman's greatest foe, his arch-enemy and opposite number - the chaos to his order. And believe it or not, it's been that way almost since the first time Joker appeared in 1940's Batman #1.

Created by Bob Kane, Jerry Robinson, and Bill Finger, the Joker was Batman's first official supervillain (though he had previously fought the Mad Monk, a vampire who wasn't considered a traditional supervillain at the time). Interestingly though, the Joker wasn't intended to stick around past that first appearance, seemingly dying in the initial version of the story. 

However, editor Whitney Ellsworth, who saw potential in the character to be a true nemesis for Batman, intervened to change the ending so the Joker would survive, through the addition of a new panel at the end of the story showing that he had escaped his apparent fate. 

The Joker then went on to appear numerous times in the Golden Age up through the early '50s, becoming Batman's number one arch-enemy not by chance, but by design of the creators of those stories.

Still, even with Joker's relationship with Batman already cemented almost from the first moment he was introduced, his characterization still didn't quite reflect the modern version of the Joker.

Back in the Golden Age, Joker was much more cunning and ruthless than some of his later incarnations. Rather than being a cackling Clown Prince of Crime, he was much more like Heath Ledger's Joker in 2008's The Dark Knight, killing civilians and causing disasters such as train derailments.

But over time, those darker elements fell out of favor not just for the character of the Joker, but for superhero comics overall, and by the mid-'40s, Joker was becoming more of a prankster, adding in more