Princess Eugenie Instagram Jack Brooksbank, Princess Eugenie and son
"We wanted to introduce you to August Philip Hawke Brooksbank," Eugenie, 30, captioned a post on Instagram on Saturday that includes three new photos of the couple and their son.
"Thank you for so many wonderful messages. Our hearts are full of love for this little human, words can't express. We are excited to be able to share these photos with you," she continued.
In choosing their son's middle name, the couple chose to honor Eugenie's grandfather, Prince Philip. The Duke of Edinburgh, 99, is currently in the hospital for "observation and rest" after "feeling unwell," Buckingham Palace said in a statement earlier this week. "The doctor is acting with an abundance of caution. The Duke remains in good spirits," a royal source told PEOPLE on Friday.
Eugenie and her sister, Princess Beatrice, are close to their grandparents, Queen Elizabeth and Prince Philip. Eugenie and Jack are now settling into life as a family of three near her grandparents at Frogmore Cottage — the Windsor home of Prince Harry and Meghan Markle. The Sussexes lent Eugenie their home, which is close to Windsor Castle, where the monarch and her husband have been staying amid the pandemic.
"Her Royal Highness Princess Eugenie was safely delivered of a son today, 9th February 2021, at 8: 55 a.m. at The Portland Hospital," the palace said in a statement. "Jack Brooksbank was present."
"The Queen, The Duke of Edinburgh, The Duke of York, Sarah, Duchess of York, and Mr. and Mrs. George Brooksbank have been informed and are delighted with the news," the statement continued. "Her Royal Highness and her child are both doing well."
The Portland Hospital in London is where Eugenie was born, as well as her older sister, Princess Beatrice. It's also where Meghan Markle gave birth to Archie, who will celebrate his second birthday in May.
Eugenie is "doing really well" after having a C-section due to her previous spinal surgery for scoliosis, a source who has spoken to the new mom since the birth tells PEOPLE in this week's issue.
The source adds that Eugenie "sounded very content and said the baby is a complete dream!"
The first photo of Princess Eugenie and Jack's baby boy was shared on the royal's Instagram page on Tuesday. The black-and-white shot shows their baby's hand wrapped around Eugenie's thumb while Jack holds his son's arm.
Eugenie, 30, captioned the shot with blue heart emojis and two exclamation points to show her excitement.
Eugenie and Jack have been quietly enjoying married life since their lavish wedding at St. George's Chapel at Windsor Castle in October 2018. On September 25, Eugenie shared their exciting baby news on Instagram with a sweet post featuring the couple holding a fluffy pair of bear slippers, writing, "Jack and I are so excited for early 2021…"
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Eugenie and Jack have been quietly enjoying married life since their lavish wedding at St. George's Chapel at Windsor Castle in October 2018.
A friend of the couple previously told PEOPLE, "Eugenie is such a fun person but very thoughtful too. I think she will be more of a bohemian parent, a bit like her mom, while Jack will be calm and chilled."