The author firstly gives the conditions such that a quartic algebraic equation has a pair of complex conjugate roots and a pair real roots, and all of the roots have strictly negative real part (Lemma 1-3). With that, he gives the conditions of coefficients of the characteristic equation which enables us to know the existence of the Hopf bifurcation. As an application, the author considers the Brusselator in diffusion. Without tedious computation, he obtains the result for existing a periodic travelling-wave solution by Hopf bifurcation.