Read His Autobiography
Read his autobiography "Willie's Game" by Willie Mosconi & Stanley Cohen. He describes what took place and keep in mind this was his second exhibition of the day, he had just driven hundreds of miles to reach his destination, he hadn't eaten lunch, and arrived late so he didn't even get to practice or learn the cloth or rails.
He only was allowed to play on a Brunswick pool table because he was under contract with Brunswick for many years, as were other touring pros, and he was prohibited from playing on any tables that weren't Brunswick tables. Ergo, the pool hall where he was performing his second exhibition that day he pocketed 526 balls, i.e., straight pool or 14:1, only had a Brunswick table in a 4'x8' size and so that was the only table he was permitted to play on despite that there other larger size pool tables at the pool hall. But since those larger tables were not Brunswick pool tables, Willie wasn't permitted to even stroke a pool ball on those tables since even hitting one ball on a pool table that was not Brunswick would have violated Willie's contract with Brunswick.
If you know much about Willie, which you will after reading his autobiography, Willie was a man of his word and the notion of breaking a contract was the last thing he'd do. But don't ever cross the man because he had a bad temper and wasn't hesitant to let it show. Anyway, if you really want to know what happened with his run of 526 balls, read his autobiography. After all, who would know better what took place and why than Willie.