I can see a transcript of a baptismal record for William, but I haven't found a civil birth registration and I'm starting to wonder if this first child went unregistered.
For what it's worth, William Douds was baptised in Richhill Presbyterian Church in September 1867 (day unspecified), date of birth given as 18 June 1867, his parents recorded as John Douds and Sarah Earls of Ballinahinch, which I assume is Ballynahinch, a neighbouring townland of Mullalelish - Ballinahinch was given when one of the couple's other children was born (Mary in 1875), Mullalelish was given for all of the rest.
On the face of it, there ought to be a registration for William in 1867 under Armagh District.