By P. C. Mahalanobis, C. R. Rao

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6) VÍ~PÍQ being the Kronecker delta. (b) Fisher (1940) showed that for a balanced incomplete block (BIB) design there holds the inequality b^v ... 8) where b is the number of blocks and ν is the number of treatments. (For an alternative proof see Bose (1949)). One may ask what the corresponding result is for P B I B designs. Now b ^ rank Ν > rank NN'. , has a zero characteristic root, in which case Π and therefore Π* has a zero characteristic root. Thus: A necessary condition for b ^ ν in a PBIB design is li>öl ^ 0 where p*^ is given by ...

H . (1963): Finite nets Π . Uniqueness and imbedding. Applications to stocheis- Pacific J. Math. CHA-NG, L . (1959) : The uniqueness and non-imiqueness of the triangular ctssociation schemes. , 8, 604-613. Science (1960): Association schemes of partially balanced designs with parameters ν = 28, η , = 12, ^2 = 15 and p]i=^. Science Record, Math. , 4, 12-18. C l a t w o r t h y , W . H . (1952) : Partially balanced incomplete block designs with two cissociate classes and three replications. University of North Carolina, Doctoral Thesis, — (1954) : A geometric configuration which is a partially balanced incomplete Proc.

C. BOSE 8. 5) and for which the inequahties 1 < Í< ^ l ^ t ^ k are satisfied is defined to be a pseudo-geometric association scheme with characteristics (r, t). , the association scheme of a partial geometry ( r , k, t). However an asso- ciation scheme may be pseudo-geometric without being the association scheme of a partial geometry (r, k, t). In particular a two class association scheme with characteristics (r, k, r—1) may be called a j95ez¿do-LY scheme. I t has the same parameters as anL^ association scheme.