404 
Proceedings of the Royal Society of Edinburgh. [Sess. 
the ordinary theorem of chance gives the distribution 
(1 +4 + 6 + 4+ l) 2 , 
and so on. 
(2) In this case, let (db) = (cc). The grouping is then as follows : — 
( aa ) + 2 (ab) + (bb) 
(cc) + 2 (cd) + (dd) 
2 (ac) 2 (be) 2 (bd) 
2 (ad) 
or 
1 + 2 + 3+ 4 + 3 + 2 +1, 
or 
(l + l + l + l) 2 . 
Again a unimodal curve shows itself. If the quality depends on n pairs of 
elements, then the general distribution is given by the multinomial, 
(1 + 2 + 3 + 4 + 3 + 2 f l) n . 
This quickly approximates to the normal curve. 
or 
or 
(3) Let (cc) fall between (aa) and (ab ) ; the grouping is then : — 
(aa) 
2 (ac) 
(cc) 
+ 2 (ab) +(bb) 
+ 2 (cd) + (dd) 
2 (cb) 2 (bd) 
2 (ad) 
1+ 2 + 1 + 0+ 2 + 4 + 2 + 0 + 1 + 2 + 1, 
(1+ I + 0 + 0 + 1 + lp, 
or we have in this case a multimodal curve. 
(4) Let (cc) fall between (ab) and (bb ) ; the grouping is then : — 
(aa) . . . 2(ab) . . . (bb) 
(cc) . . . 2 (cd) . . . (dd) 
2 (ac) . . . 2 (be) . . . 2 (bd) 
2 (ad) , 
1+0 + 0 + 2+ 2 + 0 + 1 +4+1+0 + 2 + 2 + 0 + 0 + 1. 
or 
1 + 0 + 0+1 + 1 + 0 + 0 + 1 ) 2 , 
which again ultimately approaches normality. This curve, composed of 
two nearly equal races, has three modes. 
(5) Let (cc) be greater than (bb), and we have the following, when a 
dot indicates a gap of one unit : — 
