Problem is we don't know... it could have been soliciting for sure... or.. as Mike suggested, these could have been men Tumblety had known and was just caught... or.. tried to take the "friendship" farther and they panicked, or reported him, sadly as Tumblety always had money on him, it was easy to skirt certain charges (as it is now with money), If I had to guess... and it is only a guess, Tumblety was caught trying to pick someone up, the police found the letters he had on him, busted him, but he was able to buy his way out (for the most part)... I also don't think it was a first offence as they had a file on him already.. he was a known "pervert".
The thing with Tumblety is, once he found someone, he would try to stay with them (in a casual sense) for as long as he could, as Mike has pointed out, not get close, but stay in contact (in touch would have been a bad pun) and have that kind of "relationship" .. but we know he also liked "cruising" in slums (in DC, in London, in St. Louis, in Rochester etc.)
"The truth is what is, and what should be is a fantasy. A terrible, terrible lie that someone gave to the people long ago."- Lenny Bruce