Peggy,
Death registration didn’t start in Ireland till 1864 so if Cornelius’s wife died before that (as seems likely) then there probably won’t be a record of that event.
I searched the RC parish records for a marriage for Cornelius and also for children baptised to a father named Cornelius but without success. Probably the family lived in a parish that does not have records for the period you need.
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).