Let $\beta$ be a arc length parametrization of a curve in a suface $M$. The Referral Darboux is the units vector {$T$,$V$,$U$}, where $T$ = unit vector tangent, $U$ is the unit vector normal and $V = UxT$. Show that
$T' = gV+kU$
$V' = -gT+tU$
$U' = -kT-tV$
where $k$ is a normal curvature in the direction of $M$ and $g$ is a geodetic curvature of $\beta$.
Notice first of all that $U$ and $T$ are perpendicular, so from $V=U\times T$ one also gets $U=T\times V$ and $T=V\times U$.
In addition, a unit vector is always perpendicular to its derivative, so there exist four numbers $g$, $k$, $s$ and $t$ such that: $$ T'=gV+kU,\quad U'=sT-tV. $$ It follows that $$ V'=U'\times T+U\times T'=-t(V\times T)+g(U\times V)=tU-gT, $$ and $$ U'=T'\times V+T\times V'=k(U\times V)+t(T\times U)=-kT-tV, $$ so that $s=k$.