By R. C. Bose (auth.), Prof. A. Barlotti (eds.)

R.C. Bose: Graphs and designs.- R.H. Bruck: building difficulties in finite projective spaces.- R.H.F. Denniston: Packings of PG(3,q).- J. Doyen: fresh effects on Steiner triple systems.- H. Lüneburg: Gruppen und endliche projektive Ebenen.- J.A. Thas: 4-gonal configurations.- H.P. younger: Affine triple systems.

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**Example text**

Bose and Mesner (1959) have given a general method of d c a a t i n g t h e m u l t i p l i c a t i e s of t h e r o o t s of B = NN1. We s h a l l i l l u s t r a t e t h e determination of t h e m u l t i p l i c a t i e s a,, = 1, al, a2 of t h e roots O0 = r k , el, e2, i n t h e special case m = 2. 5). Now 0 and 0 a r e t h e c h a r a c t e r i s t i c 1 2 Setting we f i n d a f t e r some calculation tinat R. C . e. parameters of the first kind. The corresponding general result is due to Bose and Mesner (1959).

E . n ij Let = 1 or 0 according a s the treatment i does or does not occur i n the j-th block. e. designs i n which w e r y treatment contrast i s estimable, HH' is irreducible. 9) t h e sum of the elements i n w e r y row of NN' is rk. Hence is a stochastic matrix ( i . e . row i s unity). For such a matrix [ ~ r a u e r(1952) 1unity is a simple Hence rk is a simple root root and i s greater than all the other roots. of B and is, therefore, also a simple root of P. One can now show hose (1963 b) 1t h a t the m characteristic roots of P other than r k a r e the roots of the matrix.

R. C. BOSE CHAPTER I11 PARTIALLY BALANCED DESIGNS ANLl ASSOCIATION SCHEMES 1. Definition of p a r t i a l l y balanced association schemes and par- t i a l l y balanced incomplete block (PBIB) designs. , o r m-th associates, t h e r e l a t i o n of association being symmetrical, i . e . , i f the treatment a i s t h e i-th associate of t h e treatment 8, then 0 i s t h e i-th associate of t h e treatment a. (b) Each treatment has ni, i-th associates, t h e number ni being independent of a. ( c ) I f any two treatments a r e i-th associates then the number of treatments which are j-th associates of a and k-th associates of 8 i s p i jk and i s independent of t h e p a i r of i-th associates a and 8.