Welcome to Rootschat. I've moved your post to the Dublin board since Arthur was born in Dublin.
To start with, he's listed in IGI (
www.familysearch.org): Arthur Hammond born 12 Oct.1873 Dublin, Co. Dublin, son of Arthur Hammond and Mary Anne Thorpe.
He's also in 1881 census on the same site-
Mary Ann HAMMOND, widow, 38, born Chesterfield, Derby, England (formerly shopkeeper)
daughter Mary Alice HAMMOND, 15, born Aldershot, Hampshire, England
son Arthur HAMMOND, 7, born Dublin, Ireland (scholar)
Address: 49 Castle Street, Farnham, Surrey, England
Index to civil registration is online (free) and gives the details needed to order the birth certificate-
http://pilot.familysearch.org/recordsearch/start.html#p=collectionDetails;t=searchable;c=1408347Arthur HAMMOND 1873, Dublin registration district, volume 17 page 569
To order the certificate see here-
www.groireland.ie/apply_for_a_cert.htm