Report Preface: From certain viewpoints intercorrelation (that is, correlation between independent variables) is not a major problem in statistical analysis. Routine instructions for solving multiple-regression problems include formulas for net regression coefficients and standard errors which automatically take account of the effects of intercorrelation. Nevertheless, research workers are frequently surprised when two analyses showing nearly the same direct correlations between the dependent and each independent variable yield widely differing net regression and multiple correlation coefficients. Many of the three-variable calculations discussed in this paper were developed by the senior author in May 1947 to explain such happenings in a precise way. Late in 1952 the junior author rechecked and extended the three-variable calculations. He also developed representative calculations for the four-variable case, setting up the intercorrelation formulas in a matrix notation which permits generalization to any number of variables. The authors are indebted to Frederick V. Waugh for helpful suggestions on the four-variable and general cases. The four-variable computations were carried out by Jacqueline Spiro.