Let \(f\) be a bounded integrable function on the interval \((-1,1)\) and \(I(f)={\int_{-1}^1 w(x)f(x) dx}\), where \(w\) is a weight function. Let further \(\varepsilon _{n}(f)=J(f)-K_{n}\), where \(J(f)=I(f):I(1)\) and\(K_{n}(f)\) is the Gaussian quadrature for \(J(f)\) with the \(n\)-knots on \((-1,1)\). For the Chebychev polynomials \(T_{m}\) of the first kind and for some Jacobi weight functions \(w\) the behaviour of the quantity \(\gamma_n(w) = \sup_{m\geq 2n}|\varepsilon_n(T_m)\|\) is investigated. This quantity has applications in determining error bounds for Gaussian quadrature of analytic functions in some ellipse with foci \(\pm 1 \).