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...Then Hermione shrieked: Harry drew his wand again and spun around to see a silver Patronus soar through the drawing room window and land upon the floor in front of them, where it solidified into the weasel that spoke with the voice of Ron’s father. “Family safe, do not reply, we are being watched.” - Deathly Hallows: Chapter 9: "A Place to Hide"
How did Arthur Weasley know where the trio was to send the Patronus to the right place? Are Patronuses self-homing, like Owls?
Yes, Patronuses seem to be self-homing, meaning the caster of the Patronus can tell it who it needs to deliver a message to, and the Patronus, like a homing pigeon, just finds the individual(s) in question.
Arthur Weasley's Patronus to Grimmauld Place is a perfect example of this. So is Professor McGonagall's in Deathly Hallows:
'We must alert the other Heads of House. You’d better put that Cloak back on.’
She marched towards the door, and as she did so she raised her wand. From the tip burst three silver cats with spectacle markings around their eyes. The Patronuses ran sleekly ahead, filling the spiral staircase with silvery lights, as Professor McGonagall, Harry and Luna hurried back down.
Deathly Hallows - page 480 - Bloomsbury - chapter 30, The Sacking of Severus Snape
(Going on the presumption that the other Heads of Houses were not in bed and may have been in other parts of the castle by this time, but some might see it otherwise)
On the other hand, we see Patronuses come through where it's clear the caster knows where the person who is to receive the message is. In Deathly Hallows, Arthur sends a Patronus to warn Molly that the Minister for Magic is coming home with him (chapter 7, The Will of Albus Dumbledore). The same happens in the next chapter of Deathly Hallows when Kingsley sends a Patronus to the wedding warning that the Death Eaters were on their way (chapter 8, The Wedding).
Snape, as well, knew Harry and Hermione were in the Forest of Dean, because Phineas Nigellus ratted them out to Snape. He didn't know where in the Forest of Dean, though. It seems a logical conclusion that Snape's doe Patronus had to know how to find Harry (chapter 19, The Silver Doe).
I conclude from the books that a Patronus can either take direction from its caster, or can find a specific place or individual through its inherent magical powers -- it just depends on the situation.
Evidence definitely supports that they are self-guided. They were taught to the Order by Dumbledore to be secretive means of communication between members of the Order of the Phoenix, therefore it is highly unlikely that they were meant to be sent to a location, rather than a specific person.
Note: JKR had said Dumbledore "taught" the Order of the Phoenix members how to use Patronus to communicate. While she doesn't say whether it was him who invented this form of communication or not I think it is implied.
15 July, 2005: So how DO the members of the Order of the Phoenix communicate with each other?
The Order communicates by Patronus; Dumbledore taught them how to do this.
The source for this is accio-quote, which is not "first hand" but the link to JKRs website has since broken. (I am assuming that it is likely on Pottermore.)
The examples mentioned thus far (in the question and answers) are possible cases of self-homing patronuses, but there is no definitive evidence in those cases. There is another instance where there is perhaps stronger evidence that patronuses can be self-homing. In Chapter Nine of Deathly Hallows we find the following dialogue:
“I say we find a quiet place to Disapparate and head for the countryside. Once we’re there, we could send a message to the Order.”
“Can you do that talking Patronus thing, then?” asked Ron.
“I’ve been practicing and I think so,” said Hermione.
“Well, as long as it doesn’t get them into trouble, though they might’ve been arrested already. God, that’s revolting,” Ron added after one sip of the foamy, grayish coffee.
Here they plan on sending a message to unspecified Order members at an unspecified location, and no one mentions anything about needing to know where the Order member(s) would be. At this point, they knew that The Burrow had just been overrun by Death Eaters, so that would probably not be the location, and they already knew that Grimmauld Place wasn't safe. That doesn't leave them many options of known locations in the first place.
Furthermore, Ron's comment about getting them into trouble would be a lot more significant if they didn't know where they were sending the messages to. If there was a possibility that the patronus would deliver the message in front of someone undesirable then there would be a real concern about getting someone in trouble; if, however, the patronus would be going to a specified secret location (such as one of the Order members' houses) then there would be less concern.
From this case it seems likely that the patronuses are not necessarily sent to a specific location, but to a specific person (or people).
On the other hand, in Chapter Eight of Half-Blood Prince we find the following:
"I meant Hagrid to get the message," said Tonks, frowning.
"Hagrid was late for the start-of-term feast, just like Potter here, so I took it instead.
This seems to indicate that the message was not sent to Hagrid, but was sent to the place where Tonks thought that Hagrid would be. If patronuses were self-homing the message could have gone to Hagrid wherever he was, instead of being intercepted by Snape.
Additionally, if patronuses are not self-homing we can easily explain various instances of wizards failing to use this method of communication. For instance in Chapter of Order of the Phoenix:
Dumbledore heaved a great sigh and then said, “Alastor Moody, Nymphadora Tonks, Kingsley Shacklebolt, and Remus Lupin were at headquarters when he made contact. All agreed to go to your aid at once. Professor Snape requested that Sirius remain behind, as he needed somebody to remain at headquarters to tell me what had happened, for I was due there at any moment. In the meantime he, Professor Snape, intended to search the forest for you.
“But Sirius did not wish to remain behind while the others went to search for you. He delegated to Kreacher the task of telling me what had happened. And so it was that when I arrived in Grimmauld Place shortly after they had all left for the Ministry, it was the elf who told me — laughing fit to burst — where Sirius had gone.”
Here we see that Snape did not send messages to Order members wherever they might be. He sent one message to Grimmauld Place, as that is the location he was aware of. Even more telling is that he had to assign Sirius to wait for Dumbledore, and Sirius in turn had to delegate that task to Kreacher. If a patronus could have found Dumbledore on its own, surely one of them would have sent him a message immediately!
If patronuses are indeed not self-homing, we would have to assume that Arthur had some other way of knowing that they were at Grimmauld Place. It might have been a lucky guess, or he might have sent a patronus to every possible location he could think of.
