v = 19, b = 57, r = 9, k = 3, λ = 1. Developing the initial blocks given here, it will be found that the blocks of second set treatments which occur with the first set treatment 1 are (written as columns):

8	2	17	10	5	8	18	17	10
18	12	8	12	7	10	7	6	18
12	6	2	7	2	5	6	5	7

Likewise, the blocks associated with first treatment 7 are:

8	14	4	16	11	14	5	4	16
18	5	14	18	13	16	13	12	5
12	18	8	13	8	11	12	11	4

while those for treatment 11 are:

8	18	12	1	15	18	9	8	1
18	9	3	3	17	1	17	16	9
12	3	16	17	12	15	16	15	8

These configurations show the plane of second set points associated with the first set points 1,7 and 11. Since first set treatments 1 and 7 occur together in a block, the configurations of second set treatments associated with them have a block in common viz (8,18,12). This block can therefore be regarded as the intersection of the two planes. Moreover, as first set treatment 11 occurs with 1 and 7 in this block, the configuration associated with 11 intersects the other two configurations in the same line. So every line of the BIBD(19,3,1) can be regarded as the intersection of 3 planes of the SBIBD(19,9,4).

There are three possible 9₃ configurations: in one, the lines can be arranged into 3 sets of 3, each forming a replicate (or sub-configuration), in another, there is just one set of 3 lines forming a replicate, and in the third there are no replicates. These are illustrated as Figures 4, 10b and 10a respectively by Vajda, so denote them by 9c, 9b, and 9a respectively. The configurations defined here are of the third type, so a suitable formula would be {9a}¹⁹.

There are 969 triples of configurations here, falling into 4 types. Using just the number of points per pattern, a simple formula would be 3⁵⁷8¹⁷¹7⁷⁴¹, where the 7-point patterns include 399 with 3 lines through a single point.

The same analysis applies when configurations of first set treatments are considered.

The triples of points and the associated triples of configurations meeting in those points, can be used to distinguish particular SBIBDs amongst others with the same parameters (see Bhat & Shrikhande). For the SBIBDs used here, the blocks that these triples form, constitute the BIBD which underlies our designs, so that there is established an intimate link between the SBIBD and the BIBD.

This example is continued here.