The abundance of manuscript material and correspondence relating to the second edition of Principia Mathematica in the Bertrand Russell Archives makes possible a reconstruction of the process of its composition. It is clear that the second edition is solely the work of Russell, and what is new is confined to the Introduction to the second edition and three appendices, with the rest of the first edition reset with corrections or simply reprinted. Frank Ramsey was involved with the new edition, reading manuscript material for the new material, and then the printer's proofs late in 1924.
From the manuscript material we can also learn more of the details of this story. The new material for the second edition was originally planned to take the form of a revised chapter of the Introduction to the first, and it survives as a manuscript of seventy odd leaves, called “The Hierarchy of Propositions and Functions.” (“HPF” in what follows.) This was the first draft, with its leaves dispersed and renumbered, later filled out with the rest of the material for the Introduction to the second edition and Appendices A and B. Appendix C appears to have been then added, with only the final draft surviving.
Appendix B, which is not reprinted in the paperback Principia Mathematica to ✻56, with the technical problems identified by Gödel (1944) and confirmed by Myhill's (1974) proof, was the subject of the most preliminary work.