For most lineage societies, you can document more than one ancestor, provided all ancestors meet the society’s qualifications. That second (or third, fourth, or fifth…) application is called a “supplemental.”
When you submit a supplemental, you can build off of your existing application and connect to the new ancestor by providing the paperwork for the generations which differed from your original line. In some cases, you may be “proving” only one new generation – if both of your ancestors’ grandfathers were in the Revolutionary War. In others, it may be much more. Beyond the fact that you may not need to document the entire line, supplementals follow the same rules as initial applications.
So, why do a supplemental? For most people, it’s a chance to record a bit more of their family history. By putting an additional line “on file,” a member ensures that their family history is documented some place it will be retained.