No. 5S3] 



\ T SUBSTRATUM HOMOGENEITY 



443 



laid out in a 20 X 25 manner. This permits only 2x5, 

 4 X 5 or 5 X 5 combinations of the same size throughout. 

 Montgomery's experiment comprises an area of 16 X 14 

 plots which may be combined in only 2 X 2 or 4 X 2 equal 

 areas suitable for calculation. In each of these cases 

 other groupings are desirable. 



The formulae are quite applicable to such cases: the 

 arithmetical routine is merely a little longer. The for- 

 mula is again 



but p and <r P are obtained by a (n — l)-fold weighting of 

 the plots, 16 where n is the number of ultimate plots in the 

 combination plot to which any p may be assigned, i, e., 



The point may be illustrated in detail on the wheat data 

 of Mercer and Hall. I adopt a combination by twos from 

 north to south, i. e., arrange the data in 10 rows of com- 

 bination plots instead of 20 rows of ultimate plots. From 

 east to west there are 25 rows of ultimate plots ; these can 

 be only reduced to a 2 X 2-fold grouping for the first 22 

 rows. The lines of division are indicated on Map B by 



Row 23-25 must be thrown into combination plots each 

 of 6 units. The possible permutations within a combina- 

 tion plot are 1/2 n(n — 1), but since the surfaces are ren- 

 dered symmetrica], the total permutations for the whole 

 field is S[n(n — 1)]. There are only two sizes of combi- 

 nation plots, of which 110 have 4 and 10 have 6 ultimate 

 plots each. Thus the weighted population N is 



{[S(C P *) -S(p*)]/S{n (n- 1)]} - p» 



p = S[(n-l)p]/S[n(n-l)], 



