1909-10.] The Significance of the Correlation Coefficient, etc. 501 
The correlations in these two cases are respectively (a) r = *578, and ( b ) 
r = - 330.* This process may be continued indefinitely, and likewise results 
may be obtained on the second hypothesis. 
If the excess of recessive be maintained for four generations the corre- 
lations are as follows : — 
A. 
B. 
Parental .... 
•578 
•638 
Grandparental 
•330 
•451 
Great-grandparental . 
•200 
•309 
Great-great-grand parental . 
•no 
•216 
But the heredity has not been quite this. The proportion of dark and 
light horses in each generation when means of each parentage are taken 
has altered in the following manner : — 
Dark. 
Light. 
Total. 
Great-great-grandparents . 
641 
359 
1000 
Great-grandparents . 
664 
336 
1000 
Grandparents 
712 
288 
1000 
Parents .... 
728 
272 
1000 
So that for two generations the proportion of recessive is one-third and 
above, and for the last two approaching the ratio of one-quarter, though, 
as before remarked, this is not of a Mendelian origin. If we calculate, then, 
the correlations for the great-great-grandparents on the hypothesis that 
the recessive is equal to one-third of the total for two generations and to 
one-quarter of the total during the next two generations, and for the grand- 
parents that on the hypothesis that the recessive numbers one-third for 
* These and the subsequent correlations have been obtained by the fourfold method 
though not by the full process. They have been calculated by the formula, 
. n 2 4a.6cdN 2 
r=sm 2 v!tP when 
and where the fourfold division is 
This formula gives results very near the truth. When those coefficients, previously calcu- 
lated in this paper by the full method, were checked by the method here referred to, the 
result has been so close that in the present instance where many coefficients are required 
the extra labour of calculation has not seemed necessary. 
