  We provide precise formulations and proofs of two theorems from Darboux's
lectures on orthogonal systems. These results provide local existence and
uniqueness of solutions to certain types of first order PDE systems where each
equation contains a single derivative for which it is solved: \[\frac{\partial
u_i}{\partial x_j}(x)=f_{ij}(x,u(x)).\] The data prescribe values for the
unknowns $u_i$ along certain hyperplanes through a given point $\bar x$.
  The first theorem applies to determined systems (the number of equations
equals the number unknowns), and a unique, local solution is obtained via
Picard iteration. While Darboux's statement of the theorem leaves unspecified
"certaines conditions de continuit\'e," it is clear from his proof that he
assumes Lipschitz continuity of the maps $f_{ij}$. On the other hand, he did
not address the regularity of the data. We provide a precise formulation and
proof of his first theorem.
  The second theorem is more involved and applies to overdetermined systems of
the same general form. Under the appropriate integrability conditions, Darboux
used his first theorem to treat the cases with two and three independent
variables. We provide a proof for any number of independent variables.
  While the systems are rather special, they do appear in applications; e.g.,
the second theorem contains the classical Frobenius theorem on overdetermined
systems as a special case. The key aspect of the proofs is that they apply to
non-analytic situations. In an analytic setup the results are covered by the
general Cartan-K\"ahler theorem.

Analysis of PDEs;Differential Geometry