The joint frequency distribution has been found for any set of the $(n - k)$ deviates from their sample mean of each of the $t$ variates in a sample from a multivariate normal population. Expressions for the variance of any single deviate in this distribution, the correlation coefficient between any pair of deviates, and certain partial coerrelation coefficients between any pair have also been obtained. These results have been generalized so as to include the corresponding properties of deviates from a set of $t$ multiple linear regression equations estimated from the sample, the $m$ independent variates being the same for each of the $t$ dependent.
"The Frequency Distribution of Deviates from Means and Regression Lines in Samples from a Multivariate Normal Population." Ann. Math. Statist. 17 (3) 344 - 349, September, 1946. https://doi.org/10.1214/aoms/1177730947