SOME CLASSES OF PARTIALLY BALANCED DESIGNS

1. Summary. Incomplete block designs with a few replications are of practical interest to experimenters. Partially balanced incomplete block (PBIB) designs wvith two associate classes, and k > r = 2, were studied by one of the authors [11. The present paper extends this investigation to the case kc > r with XI = 1 and X2 = 0. It is shown that the parameters of all PBIB designs in this case are given by (4.27) anid thus depend upon three integral parameters k, r, and t, with the additional restrictions that (i) 1 ? t r is 5. However designs with parameters (4.27) with r = 3, t = 1, and k = 2 or 3 are also combinatorially possible though they do not belong to the class k > r. Interesting by-products of this study are a lemma and five corollaries which give an insight into the structure of PBIB designs with Xi = 1 and X2 = 0, and pli = k - 2, no special assumptions being made regarding r and k.