It is an obsolete process, now that birth certificates are issued by the County Clerks office, therefore many clerks would not know about delayed births. This was done by the Probate Judge when an adult did not have a certificate of birth. The individual affected would provide a proof and the Probate Judge of record would determine if the facts warranted an actual certificate. My father had two living sisters who testified under oath when they recalled their brother being born in their house. The Judge was satisfied that these sisters provided information to support the birth. My father needed a birth certificate for World War II purposes and many other people needed them for retirement and Social Security purposes. The important thing was that the applicant would present someone or something before the judge to establish birth. It had to be a court of record with a presiding judge. The actual document would then be filed at the county clerk's office for posterity. There were many births that were not recorded in a timely fashion and simply fell through the cracks.