the best value of the coupling-ratio from a given set of data. 439 
deduced from some assumed ratio—as in the case of the above 
data for peas and the ratio 15 : 1—1s very slight evidence in favour 
of the truth of the assumption, especially where the coupling-ratio 
is high, at least with such moderate numbers of observations as 
are at present available. Some light might, however, be thrown 
on the theory of reduplication by carrying out an examination of 
all the available cases, determining p or the coupling-ratio for each 
by equation (1). Such an examination we hope to carry out. 
As p is not expressed explicitly as a function of the propor- 
tionate frequencies f by equation (1), we do not see our way to 
give its probable error by this method of determination. The 
value given by (2), however, is in some cases close to the value 
given by (1), viz. if no one of the frequencies is very small (cf. the 
data below), and its standard error can be determined without 
difficulty on the usual, though hardly quite justifiable, assumption 
that deviations in the frequencies are small compared with their 
mean values. As the standard errors by the two methods of 
determination are likely to be of the same order of magnitude, it 
seems worth while stating the result as at least a rough guide to 
the possible magnitude of fluctuations. Ditferentiating both sides 
of equation (2), squaring and summing, we have, utilising known 
results for the sums of squares and product sums (cf. e.g. Yule, Jl. 
Stat. Soc. 1912, p. 601), 
ae) ees , 
where ¢, is the standard error of p (to be multiplied by 0°6745 to 
obtain the probable error) and JN is the number of observations. 
If there is coupling (p > 0°25), the coupling-ratio r = p/(0°5 — p). 
Differentiating, squaring and summing again, we have 
CH= oS F cupaddcOnaoddOODODOCO000D (4). 
Standard error of the 
Neinber Value of p from r from values from (2) 
of obser- 
vations 
Case (1) (2) (1) (2) p r 
Wheat 213 0-1404 | 0-1968 || 2:5 
Maize | 2736 || 0°3891 | 0°3885 || 3-5 
Peas 885 0:4745 | 0:4744 || 18-6 
2°56 | 1:54 || 0:0430 0:56 
1; 3-48}/ 0:0049 | 0:20 
18-5 || 0°00385 , 2°94 
If there is repulsion (p< 0°25), the repulsion-ratio is (0°5 —p)/p 
and p* must be read for (0°5 — p)‘ in the denominator of the above 
