The latin square design has been extensively used in field experiments, because of its ability to eliminate from the experimental errors the effects of soil fertility variations in two directions. More recently, Yates (1, 2) developed the lattice square designs for varietal trials conducted under a plant breeding program. In these designs, the number of varieties must be an exact square. Within each replication, the plots are laid out in a square array, in which every row and column contains the same number of plots. (The physical dimensions of the rows and columns will, of course, be different unless the plots are square in shape.) In successive replications the groupings of the varieties into rows and columns are changed so that with the complete design every pair of varieties has appeared together once either in the same row or in the same column. This symmetry makes it possible to adjust the varietal total or average yields, by simple calculations, for variations in the fertility of different rows and columns. In this way the effects of soil fertility variations in two directions can be eliminated from the experimental errors, just as in the latin square.

These designs will be described as balanced lattice squares, in order to distinguish them from the designs which form the subject of this bulletin. In the useful range for most practical purposes, the available selection of balanced lattice squares is shown in table 1a.