Let \(B_{n}(x)\) be the \(n\)-th term of the conjugate Fourier series of \(f(x),\) and let \(\psi(t)=f(x+t)-f(x-t)-l\). The author shows that if \[ \Psi(t)=\int_{0}^{t}\psi(u)du=o(t) \text{ and } \int_{\varepsilon}^{\delta}\frac{|\psi(t+\epsilon)-\psi(t)|}{t}dt\rightarrow 0 \] as \(\varepsilon\rightarrow 0\) for some fixed \(\delta\), then the sequence \(\{nB_{n}(x)\}\) is summable \((C, 1)\) to the value \( l/\pi\). The method of proof is very near to that of the \((C, 1)\) summability of the derived Fourier series (cf. [\textit{A. Zygmund}, Bull. Acad. Polon. 1925, 207--217 (1926; JFM 52.0271.04)]. From this he derives Lebesgue's test for convergence of the conjugate Fourier series by applying Tauber's theorem.