David Cameron gave the baby a hasty kiss on the forehead and then indicated that he wanted to hold him. Mrs Cameron stood with a glazed look and fixed smile as the Tory leader took the infant from her.
The choice of an old Celtic name has no special significance, Mr Cameron said, although he added that there might have been another Arthur Cameron in the family tree.
"Arthur we've always liked as a name and Samantha found Elwen in a book and we thought it was a good name," he said.
The Cameron's third child was born at 11.55am on Tuesday in the Clarence Memorial ward of St Mary's Hospital, Paddington, weighing 7lb 13oz. His arrival has delighted not only his parents, but also the professionals in charge of modernising the image of the Conservative Party.
They were driven almost to despair six years ago, when Tony and Cherie Blair produced Leo, while the then Tory leader, William Hague, and his wife Ffyon remained resolutely childless. Now the Conservatives have a perfect counterweight to the child Gordon and Sarah Brown are expecting in July.
The three-day delay in revealing little Arthur's name also guaranteed the Cameron's a second batch of family friendly headlines in one week - evidence that those who advise the Tory leader on handling press know their job. However, the photoshoot itself undoubtedly lacked their usual sure touch.
Mr Cameron has been criticised for voting against new legislation giving fathers of new-born children paternity leave, and then awarded himself two weeks' leave from his front line political duties to be with little Arthur and his older brother and sister.
But he emphasised: "I'm not taking two weeks off, I've probably signed about 100 letters already this morning. I'm still working and my constituents will still have a full service."
He emerged from the house, dressed casually, to go to the shops several times, and at lunchtime he took Ivan, aged three, and Nancy, aged two, to the park.
He said that despite sleep deprivation the family were doing "fine".
"It's absolutely great to be home and it's actually been very easy. Arthur is doing really well. He is feeding nicely and doing all the right things. He was wide awake before we came out [before the cameras] and then he suddenly decided to shut his eyes."
He said Ivan and Nancy were also getting used to the idea of a new baby brother.
"They seem very chuffed about everything and it's lovely. I have had better nights' sleep but that's all part of it."
Later, the whole family set out for Mr Cameron's Witney constituency, in Oxfordshire.Reuse content