Re: Jesse Adair, military record, Bunker Hill

Family correspondence about my ancestor Jesse Adair, Royal Marines indicated that he was one of the first on the Redoubt at Bunker Hill.I have found supporting material for this in: Clark, W.B. (ed.) 1964.Naval Documents of The American Revolution. Vol. 1, Washington, p. 731 which gives a letter 21 June 1775"Major John Tupper, Royal Marines, to John Montagu, Lord Sandwich
My LordThe 17th. Inst. part of the Troops at Boston to the Number of about two thousand being Order'd to attack a Redoubt and the very Strong holds which the Rebels had in this place, the First Battalion of Marines being Included, Major [john] Pitcairn took the Command and Order'd me from the 2d. Battalion to Serve under him on this Occasion.I have now the honor to inform your Lordship that the Officers and Men of the Marine Corps shewed on this Occasion the greatest Interpidity and Valour, -----.
I take the Liberty to recommend to your Lordship favor Lieut. Jesse Adair who went with us as a Volunteer; as belong'd to the 2d Battalion, he was as I am inform'd the first who got upon the Redoubt and Shew'd the Example to others to follow him.he is the Eldest Lieutenant here.Lieut & Adjutant Waller distinguished himself also in a particular Manner; I must also recommend to your Lordship favor Mr. Thomas Simms and Mr. Samuel Davys Bowman both Volunteers the latter was Wounded.
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I have the honor to be My Lord [&c.]Jne Tupper
Charles Town Hill Camp near [Boston] 21st June 1775."

A footnote to the letter gives:PRO, Admiralty 1/486