When fitting the first-degree Scheffé polynomial and the assumption of homogeneous error variance is suspect, one suggestion made in this paper is to place the design points in a symmetrical arrangement and along the component axes. For if the magnitude of the error variance (or observation variance) follows one of several conceivable symmetrical patterns, the use of the symmetrical axial designs will result in the existence of certain relationships between the parameter estimates obtained using an unweighted analysis versus using a weighted analysis. Some results which carry over to the parameter estimates in a second-degree polynomial forq= 3, 4 and 5 are presented along with two real examples.