We give some relationships among the intersection numbers of a distance-regular graph \(\Gamma\) which contains a circuit \((u_1,u_2,u_3,u_4)\) with \(\partial(u_1,u_2) = 1\) and \(\partial(u_2,u_4) = 2\). As an application, we obtain an upper bound of the diameter of \(\Gamma\) when \(k \geq 2b_1\).