Death certificates can reveal a tremendous amount of information about your ancestors but they need not be the final piece of the jigsaw. If you are lucky then an obituary may appear or you may find a will but even without either of these things, there is a great way to look for further information.
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So what information can you find? It will vary even within the same hospital - from the briefest of details perhaps regarding your ancestor's admission to hospital to full clinical notes or a post mortem. The great news though is that even with the briefest of details, these can easily reveal some interesting facts and open up a new way to investigate a branch of your tree.
A good example of this was Edinburgh Royal Infirmary who on admission to the hospital, asked patients for their "native place" or in other words, their place of birth. This can be invaluable for those seeking to place their Irish ancestors as it would often name a town or at least a county.
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Hospitals were not always in the immediate area in which your ancestors lived so if your relative died in hospital, you may find it is some distance from their home. This is usually because this was a hospital that was free or perhaps, the one to which their workplace or church subscribed.
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They are not usually able to look for someone without details of when they were in the hospital as the information is not in a nice modern database where they can simply search for someone's name!
Also, think closely about what information you would like to find out. The archivists will answer your questions but they cannot answer questions that you have not asked!