Not sure why you've put this in the Cornwall lookups; you can find William Ball's family in 1881 and 1901 in St. Helier, Arthur A. Ball is with them in 1881 - he joined the Navy, right? Those records are up on documentsonline if you don't have them already.
In 1901 some of the children (that is, Arthur's siblings) have married and are living in Jersey - Edward J. Ball who is 12 in 1881 is aged 31 in 1901, b. St. Helier but living in St Peter Port in 1901 with wife Lena and two daughters. Alfred Ball aged 10 in 1881 is still in St. Helier, aged 30, with wife Angeline, no kids.
www.bt.com - there is a Catherine M. Ball who now lives at Great Union Road in St. Helier! Maybe a coincidence, I know, but a neat coincidence anyway.
http://www.jerseyheritagetrust.org/ - might also want to try these folks for old photos.