Where can I find a proof of Dupin's Theorem? I'm trying find a proof of Dupin's Theorem. It claims:

The surfaces of a triply orthogonal system intersect each other irr lines of curvature. (Three families of surfaces are said to form a triply orthogonal system in an open set $U \subset \mathbb R^3$ if a unique surface of each family passes through each point $p \in U$ and if the three surfaces that pass through $p$ are pairwise orthogonal.)

Thanks for all hints.
 A: I had fined it in these books:
Differential Geometry of Curves and Surfaces [Manfredo P. do Carmo]
Problem 20 Page 152 Hint 484
Geometric Methods and Applications [Jean Gallier][2ed]
Theorem 20.3. Page 629
Here is an sketch of the proof:(from second book) First,we note that if two surfaces $X_1$ and $X_2$ intersect along a curve $C$, and if they form a constant angle along $C$, then the geodesic torsion $\tau^1_g$ of $C$ on $X_1$ is equal to the geodesic torsion $\tau^2_g$ of $C$ on $X_2$ .Indeed, if $θ_1$ is the angle between $N_1$ and $n$, and $θ_2$ is the angle between $N_2$ and $n$, where $N_1$ is the normal to $X_1$, $N_2$ is the normal to $X_2$, and $n$ is the principal normal to $C$, then $$λ:=θ_1−θ_2,$$ where $λ$ is some constant,and thus $$\frac{dθ_1}{ds}=\frac{dθ_2}{ds}$$, which shows that $$τ^1_g=τ−\frac{dθ_1}{ds}=τ−\frac{dθ_2}{ds}=τ^2_g$$. Now, if the system of surfaces is triply orthogonal, letting $τ_{ij}$ be the geodesic torsion of the curve of intersection $C_{ij}$ between $Xi ∈F_i$ and $X_j ∈F_j$ (where $1≤i<j≤3$),which is well defined,since $X_i$ and $X_j$ intersect orthogonally, from an easy observation (exercise 19 from first book) the geodesic torsions of orthogonal curves are opposite,and thus $$τ_{12}=−τ_{13}, τ_{23}=−τ_{12}, τ_{13}=−τ_{23},$$ from which we get that $$τ_{12}=τ_{23}=τ_{13}=0.$$



*

*Problem 19: Let $C\subset S$ be a regular curve in $S$. Let $p \in C$ and $\alpha(s)$ be a parametrization of $C$ in $p$ by arc length so that $\alpha(0) = p$. Choose in $T_p(S)$ an orthonormal positive basis $\{t, h\}$, where $t = \alpha'(0)$. The geodesic torsion $\tau_g$,  of $C\subset S$ at p is defined by $$\tau_g=<\frac{dN}{ds}(0),h>$$
Prove that
a) $\tau_g = (k_1 - k_2) \cos(\phi)\sin(\phi)$, where $\phi$ is the angle from $e_1$ to $t$.

