By looking at Coolidge's "Algebraic Plane Curves" Ch. VII, one may guess that $p$ stands for Plücker. You should have a look at the reference cited by Coolidge in his footnote to the first page of Ch. VII with title "Plücker's equations and Klein's equation" where the notion of genus is presented. The footnote says "For an historical account, see Berzolari, p. 343". The citation is to:
Berzolari, `Allgemeine Theorie des höheren ebenen algebraischen Kurven', in Enzyklopädie des Math. Wissenschaften, vol. iii, Part $2^1$, Leipzig, 1906, 99.