According to the College of Arms there is no such thing as a coat of arms for a surname. Some people of the same surname may be entitled to different coats of arms and many may not be entitled to coats of arms. For a person to have the right to a coat of arms they must have had it granted to them or be descended in the legitimate male line from a person to whom arms were granted or confirmed in the past.
In 1623 the Heralds visited Gloucestershire to list those who were entitled to a coat of arms. The book compiled from their visit does not include any one with the surname Adey or anything similar.
A Collection of Coats of Arms borne by the Nobility and Gentry of the County of Glocester edited by Sir George Nayler and published in 1792 includes the Adey family of Wotton Under Edge, the Ady family of Dursley and the Adye family of Cirencester.

Family tradition claims that arms were granted to the family by the Black Prince at Calais. In 1615 John Adey used the arms granted to the Adye family of Kent. These consist of a blue shield with a broad golden strip across it bent five times. Two silver cherubs with golden wings are shown above the strip and one below. According to Sir George Naylers book these are the same as those belonging to the Adye family of Cirencester. The introduction to Sir George Naylers book states that the information used had not been checked for accuracy and information provided by the College of Arms and other sources does not agree with his information.
According to the College of Arms the designs shown above as belonging to the families from Wotton-under-Edge and Dursley actually belong to the Hadey family (pre 1530) about whom nothing is known. The design shown as belonging to the Adye family of Cirencester actually belonged to the Adye family of Kent and Southwark. The original hand written book of the Visitation of the Heralds to Kent by Edward Bysshe in 1668 includes this family together with a short pedigree, a drawing of the coat of arms, and the signature of Edward Adye.
In 1898 Robert Hovenden F.S.A. produced a book containing information from The Visitation of Kent from 1619-1621 by John Philipot, Rouge Dragon and William Camden, Clarenceux which included a more detailed pedigree. The accuracy of this book is not known.
The information at the College of Arms shows that the right to bear arms ceased in the early 1700s when there was no male heir. Mary Adye married Henry Stevens and the Adye coat of arms was absorbed into the Stevens coat of arms as one quarter of the design.
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e-mail to: john@adey.org.uk