If the stick were found he would say that he had left it at the office on the Saturday, and that the clerk must have borrowed it.

There was nothing unlikely in that: it was a good reason, it would explain why it came to be found near the body.
Naturally, the police would believe the word of the Mayor: it would be a queer thing if they didn't, in Mallalieu's opinion.

And therewith he tried to go to sleep, and made a miserable failure of it.
As he lay tossing and groaning in his comfortable bed that night, Mallalieu thought over many things.

How had Stoner acquired his information?
Did anybody else know what Stoner knew?
After much reflection he decided that nobody but Stoner did know.

Further reckoning up of matters gave him a theory as to how Stoner had got to know.