If you do find an entry in the Adopted Childrens Index, you can't just order the certificate with the reference numbers as you would with a birth, marriage or death.
You have to give the adopted name, and the date of birth which isn't shown in the index. All adopted children are given a short form birth cert, if you have a copy of this you will know the date of birth.
I think you may also have to give the names of the adoptive parents too as you do with a recent birth.
Dawn