Let \((S, +)\) be a commutative semigroup, \(\sigma : S\to S\) be a homomorphism such that \(\sigma \circ \sigma= I\) and \((H, +)\) a uniquely \(2\)-divisible commutative group. The author studies the additive Cauchy equation \(f(x + y) = f(x) + f(y)\) and the quadratic functional equation \(g(x + y) + g(x + \sigma(y)) = 2g(x) + 2g(y)\) for all \(x, y\in S\), and finds the function \(g: S\to H\) that satisfies the following functional equation: \[g(x + y + z) + g(x + \sigma(y)) + g(y + \sigma(z)) = g(x + \sigma(y) + z) + g(x + y) + g(y + z).\] Further, he solves the functional equation \[f(x + y) + g(x + y) + g(x + \sigma(y)) = f(x) + f(y) + 2g(x) + 2g(y)\] for functions \(f, g: S\to H\) for all \(x, y\in S\).