 Let \(\Sigma^1_{1,m}\) denote the class of \(\Sigma^1_1\)-formulas \(\exists X_1\ldots\exists X_k\phi\), where the bound second-order variables \(X_i\) are at most m-ary and \(\phi\) is first order. Similarly \(\Pi^1_{1,m}\) and \(\Delta^1_{1,m}\).