From AMS Glossary
The correlation between the residuals of two random variables (variates) with respect to common regressors.
Denoting the regression function of two variates y and z with respect to a common set of regressors x1, x2, · · · xn by Y and Z, the coefficient of partial correlation between y and z is defined as the coefficient of simple linear correlation between (y - Y) and (z - Z). To estimate the partial correlation, it is usually necessary to resort to sample approximations Y′ and Z′ of Y and Z. In that case, the estimate of the partial correlation is the sample value of the coefficient of simple, linear correlation between (y - Y′) and (z - Z′). In the simplest case in which Y′ and Z′ are taken as linear functions of a single variable x, the sample estimate ryz.x of the partial correlation coefficient is given by the formula
where the symbol ruv denotes the sample coefficient of linear correlation between any pair of variates u, v.