In the application of best linear unbiased prediction to evaluation of dairy sires, computational considerations have required that relationships among cows and their descendants be ignored. Hence, a sire's evaluation does not incorporate the information from all of his female relatives, nor does it reflect information about the dams of his daughters. A procedure which uses this information becomes computationally feasible if some relationships are approximated. This procedure is equivalent to computing under a model which includes both sire and maternal grandsires. © 1979, American Dairy Science Association. All rights reserved.