His gt-gt-gt-gt-grandson William became the first baronet of Preston but, because he had no children, and his brother had only daughters, the baronetcy was created with a remainder to his heir male.
This baronetcy devolved on the even more distant descendant of his brother John, William Hamilton, professor of logic and metaphysics at Edinburgh University.  William proceeded to prove to the courts that he was heir male and then assumed, with no conflict, the baronetcy, which his descendants then maintained.  The advantage of this court case is that strong evidence must have been presented and therefore stored in the court records to justify the detailed family history in Burke's Peerage and Baronetage.  A task for the future is to uncover the court records and better account for the long descent from John Hamilton to the Balfours and then to the Browns and then to the Trotters of whom one was my grandmother.

TFPL: Nov 5, 2002:
Diane Baptie has located the following:
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Service of Heirs
When someone fell heir to landed property, they would often apply to Chancery to be recognised as that heir.  A jury would then be empanelled and their verdict in the form of a Service or Retour would be issued.  After that, the heir would take title to the property and this would be recorded in the Particular Register of Sasines for the county in which the lands lay.  Retours are generally not very informative and if the heir was decerned as heir _general‘ the actual property is not named.  They are also in Latin until 1847

C22/109 - Retours (Services of Heirs)

22 August 1816, no.22

The jury consisted of advocates, writers to the signet and the odd _residenter‘ in Edinburgh and the inquest was held before the Sheriff depute.  The actual descent is described as follows:

'Robertus Hamilton de Preston et de Fingalton Eques Baronettis (Frater Gulielmus Hamilton de Preston Equitis Baronetti) filius Thoma Hamilton de Preston Militis filii Roberti Hamilton filii Georgii Hamilton de Preston Militis filii Davidis Hamilton de Preston Militis filii Roberti Hamilton de Preston Militis filii Domini Roberti Hamilton de Preston et de Fingalton Militis filius atnepotis Abavi Tritavi Gulielmi Hamilton Armigeri Advocatis Latoris praesentium filio Gulielmi Hamilton Anatomices et Botanices in Academia Glaguensis Professoris filii Thoma Hamilton in eodem Academia Anatomices et Botanices Professoris filii natu quarto Gulielmi Hamilton de Airdrie in Vicecomitatu de Lanark, filii Roberti Hamilton de Airdrie filii Gavine Hamilton de Airdrie filii Joannes Hamilton de Airdrie filii Gavine Hamitlon de Airdrie filii Methusalae Hamilton de Airdrie filii Joannis Hamilton de Airdrie filii natu secindi dict. Domini Roberti Hamilton de Preston et ac Fingalton Dict. Gulielmus Hamilton Armiger Advocatis Lator praesentium est propinquor et legitimus Haeres Masculis in generali‘
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The last bit is in three parts.  First it traces the ancestry of the last recognised baronet, Robert Hamilton, to his forbear Sir Robert Hamilton of Preston and Fingalton.  Second it traces the ancestry of the claimant, William Hamilton ("Armigeri Advocatis Latoris") to the same Sir Robert Hamilton of Preston, etc. Finally it asserts that said William is the heir male in general of said Sir Robert.

This was accepted by the court, but we have no real evidence of the claim.

At least the court statement is the same as that in Burke's peerage and baronetage of 1885!
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A Jonet Hamilton was the daughter of a Robert Hamilton of Preston, says The House of Hamilton.  She married around 1520, and died in 1553, a John Mamilton of Broomhill who died in 1526  after they had had about 7 children so the marriage was about 1515-1520.  These dates make it likely that this Robert was her father.

It looks like two of Jonet’s daughter married ancestors of the hamilton of Barnes and so are ancestors of the the Claud Brown who took on his mother’s name of Hamilton, of Barnes.  and those Brows were first cousins of a gt-gt-grandfather of mine.  (Phew!)

And the House of Hamilrton also confirms that it was indeed this Robert who was the father of Janet/Jonet.  See5 p. 689.
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