Unfortunately, as he had three years left at Trinity College, Hamilton was not in a position to propose marriage. However Hamilton was making remarkable progress for an undergraduate and submitted his first paper to the Royal Irish Academy before the end of 1824, which was entitled On Caustics.
Sa kasamaang palad, bilang niya tatlong taon na natitira sa Trinity College, Hamilton ay hindi sa isang posisyon sa iminumungkahi ng kasal. Subalit Hamilton ay paggawa ng pansin-unlad para sa isang undergraduate at naisumite ang kanyang unang papel sa Royal Irish Academy bago ang katapusan ng 1824, kung saan ay may karapatan Sa Caustics.