Jordan showed that the incidence variety of a smooth cubic surface containing 27 lines has solvable Galois group over the incidence variety of a smooth cubic surface containing 3 skew lines. As noted by Harris, it follows that for any smooth cubic surface, there exist formulas for all 27 lines in terms of any 3 skew lines. In response to a question of Farb, we compute these formulas explicitly. We also discuss how these formulas relate to Schläfli's count of lines on real smooth cubic surfaces.

We thank Benson Farb for asking this paper's motivating question. We also thank Matt Baker, Dan Margalit, Joe Rabinoff, Bernd Sturmfels, and Jesse Wolfson for helpful suggestions and support. We thank the anonymous referee for their detailed comments and suggestions that greatly improved the clarity and accuracy of the paper. Finally, we are especially grateful to Steve Trettel for the included graphics.