  The sixth Painlev\'e equation is a basic equation among the non-linear
differential equations with three fixed singularities, corresponding to Gauss's
hypergeometric differential equation among the linear differential equations.
It is known that 2nd order Fuchsian differential equations with three singular
points are reduced to the hypergeometric differential equations. Similarly, for
nonlinear differential equations, we would like to determine the equation from
the local behavior around the three singularities. In this paper, the sixth
Painlev\'e equation is derived by imposing the condition that it is of type (H)
at each three singular points for the homogeneous quadratic 4th-order
differential equation.
