Hello,
Have you tried freebmd? for the birth registration index, with that you can apply for a birth certificate.
A quick search I did revealed quite a few Petty's, one James Ashley Petty, District Southampton, Vol 2c, Page 41, March 1879. If he was born in December 1878 its quite possible the birth wouldn't be registered until January 1879 - thus the March quarter.
Freebmd is not complete though so there could be others, there's no way of telling from the index if it's the right one, but when you apply for the certificate you can give other reference points, e.g. fathers name is .......
then if it' s not the right one they don't charge the full amount, or apply to the Registry Office for the District, sometimes they don't charge at all if it's not found - or so I understand.
Good luck
Newbie