Just from my experience, the best first approach to questions of firstness, secondness, thirdness, and so on is to regard k-ness as the property that all k-adic relations possess in common. There is more to say once this first point is appreciated but it is critical to begin from this understanding.
It is best to view k-adic relations as whole sets of k-tuples rather than fixating on single k-tuples at a time since all the most relevant properties of relations are “holistic” properties of whole sets or whole systems that are not reducible to properties of their individual elements.
A k-adic relation and its converses, numbering k! possibilities in all, each bears essentially the same information about the relation of its domains. This means that fixating on a particular ordering will tend to distract us with inessential features of a particular presentation rather than highlighting the essential properties of the relation in view.
Peirce demonstrates in several places that he appreciates the significance of these facts.
Just my k bits …
cc: Peirce List