This translation by Jim Glazebrook completes Volume.2 of Valiron's Cours d'Analyse. Refer to the Preface of Lie Groups, Volume XIV, for my reasons for undertaking this project. As pure and applied differential geometry gets ever fancier and more complicated, it is well to keep in mind where it all came from: Darboux's Theory des Surfaces. However, there is nowhere in the English-language literature where a student can find an adequate explanation of this "classical" point of view. (The only counterexamples to this statement are Eisenhart's treatises. But they are a pale rendering of the original.) When I was a student in the 1950s, I found Valiron's distillation of this material to be most helpful. I hope Glazebrook's translations will help the present generation to rediscover this Mother Lode of geometric wisdom.
This is probably the last of the series of Translations. I started over ten years ago with the seminal papers of Lie, Ricci, and Levi-Civita, and have fulfilled the goal that I set myself: To make available in English