Summary: We investigated the oscillation of the solution of the second-order mixed partial difference equation with two constant coefficients \(p{U_{m + 2, n}} + p{U_{m, n + 2}} - {U_{m, n}} + {U_{m + \sigma, n - \tau}} = 0\) by using the envelope theory, and obtained three necessary and sufficient conditions for the oscillation of solutions of the equation, where \(\sigma \) and \(\tau \) are positive integers, \(m\) and \(n\) are non-negative integers, \(p\) and \(q\) are real numbers.