Barnes, it seems, is to go immediately on his arrival, at whatever hour, to the Hotel Francis.
Presumably he would know from whom the message came, and the sender does not seem to have doubted that if it was conveyed to Barnes he would obey the summons.

Wrayson agrees to and does deliver it.

That is to say, he writes it down and leaves it in the letter-box of Barnes door, Barnes not having yet returned.

Now we begin to get mysterious.

That communication from our friend here has not been discovered.