Let Q denote the class of quasiconformal mappings of the unit disk D onto itself. For \(\kappa \in L^{\infty}(D)\), \(\| \kappa \|_{\infty}=k<1\), let \(f^{\kappa}\in Q\) denote the mapping with complex dilatation \(\kappa\), normalized by \(f^{\kappa}(0)=0\), \(f^{\kappa}(1)=1\). It is known that it is possible to interpolate between \(w=f^{\kappa}(z)\) and the identity mapping by normalized intermediate mappings \(w=f(z,t)\), \(0\leq t\leq T=\log [(1+k)/(1-k)]\), satisfying a Löwner-like differential equation \[ \frac{dw}{dt}=F(w,t),\quad w(0)=z, \] in such a manner that \(\frac{\partial F}{\partial \bar w}\) is bounded. The boundedness of \(\frac{\partial F}{\partial \bar w}\) can be achieved, for instance, by defining f(z,t) as \(f^{\mu}\), where \(\mu =(\tanh t)\kappa (z)/k\). Suppose now \(F^*(w,t)\) is defined by \[ F^*(w,t)=\frac{1}{2\pi}(1- | w|^ 2)^ 3\int_{| \zeta | =1}\frac{F(\zeta,t)}{(1- \bar w\zeta)^ 2| w-\zeta |^ 2}| d\zeta |. \] According to Theorem 4.1 of the paper, the differential equation \[ \frac{dw}{dt}=F^*(w,t),w(0)=z, \] has a unique solution \(w=f^*(z,t)\), \(0\leq t\leq T\). Moreover, \(f^*(z,t)\) belongs to Q for each t, and has the same boundary values on \(\{| z| =1\}\) as f(z,t). Also, \(f^*\) satisfies certain invariance properties with respect to composition with Möbius transformations. By means of elementary estimates and an elementary distortion theorem for Q it is proved (Theorem 6.1) that if \[ \iint_{D}\kappa (z)z^ ndxdy=0 \] for all non-negative integers n then \(f^*(z,t)\) is \([1+o(t)]-qc\), (t\(\to 0)\). With the help of known techniques this provides another proof of Hamilton's necessary condition for a mapping of class Q to have minimal maximal dilatation in its homotopy class with fixed boundary values. [The disk D can be replaced by an arbitrary Riemann surface whose universal covering surface is of hyperbolic type.]