The important problem of Voiculescu’s free probability calculus is to find the probability distribution \(\mu\) of the sum \(X_ 1+ X_ 2\) of two freely independent random variables, given the distributions \(\mu_ 1\) and \(\mu_ 2\) of the summands. Here, \(X_ 1\) and \(X_ 2\) are linear operators on a Hilbert space, and the distributions are taken with respect to a fixed vector state. In the case of bounded random variables, the above problem was solved by D. Voiculescu in J. Funct. Anal. 66, 323-346 (1986; Zbl 0651.46063). The author relaxes the assumptions on \(X_ i\), considering (self-adjoint) variables of finite variance. The operator \(X_ 1+ X_ 2\) is then essentially self-adjoint and the distribution \(\mu\) of its closure is \(\mu_ 1\boxplus \mu_ 2\), a measure whose reciprocal Cauchy transform is \(F_ 1\boxplus F_ 2\) – the free convolution product of the reciprocal Cauchy transforms \(F_ 1\), \(F_ 2\) of the distributions \(\mu_ 1\), \(\mu_ 2\). A beautiful geometric definition of the free convolution product is preceded by a careful examination of the properties of the reciprocal Cauchy transform. A free central limit theorem and the Lévy-Khinchin formula, corresponding to the case considered, are also proved.