Beginning students of analysis of variance frequently find that the explanations concerning numbers of degrees of freedom are rather more mystical than explanatory, and that the dependencies and independencies among parameters and linear restraints are not systematically covered or explicitly presented. It is the purpose of this paper to set forth the simple matrix representation of these models in a way which has been found useful for teaching purposes, and may be found of some value in the setting up of mathematical models for experimental designs calling for the analysis of variance or the analysis of covariance. The system I shall portray is a more or less natural outgrowth of suggestions made by Dr. R. C. Bose, with whom I have discussed some of the features very slightly. I am also indebted to Professor Frederick Mosteller for assistance with the mechanics of some of the models. Any deficiencies which the system may have are not to be laid to the doors of these gentlemen, however, as they have not had any opportunity to go over the material at any length. I should say also that there is no new contribution to theory involved. And I cannot assert that there is any new contribution of any other sort, because I have not corresponded with enough people on the subject to be sure that the whole system does not already exist in some publication or another. And I have not found the time to make any exhaustive search of the literature. All I can say is that I haven't seen the method presented anywhere. I have been told that Professor Henry Mann of the Ohio State University uses some such system in his teaching. This information came through Mr. David Bakan of Ohio State, and unfortunately my request for a copy of Professor Mann's materials y~elded the reply that he had no copy of his work at hand to sen4 me. In order to simplify the presentation I shall use throughout the variance ratio test, employing Snedecor's F in all cases, instead of going to the trouble of converting to natural logarithms for the Fish-