Apologies, I omitted a crucial part of the procedure.
If the father's name is not on the birth certificate, then in order to record the legitimation of the child by the parents' subsequent marriage, then they had to produce not only the marriage certificate but also a sheriff's warrant as authority to the Registrar to create the entry in the Register of Corrected Entries.
I imagine that this would involve considerable expense as the sheriff is required to make various enquiries before granting the warrant; in particular he has to satisfy himself that both parents were free to marry at the time of the child's conception. Not an easy procedure in the 1870s - how would the Sheriff be able to check that neither party was previously married if they deliberately tried to conceal that from him?