390 



Prof. Karl Pearson. 



for first parent and offspring are very nearly equal ;* assuming 

 the equality of all possible r/s for the sth parent and offspring, and 

 neglecting e we have 



2, = <r,/2*1 



> to- 



P P - 2%r p J 



or, we conclude that very approximately : The standard deviation of 

 the mid- sth parent may be obtained from the standard deviation of indi- 

 vidual sth parents by dividing by 2? s , and the correlation between mid- 

 sth parents and mid (s + p)th parents may be obtained by multiplying 

 the correlation between an individual and any sth parent by 2-P, 



Thus the variability of the ;sth mid-parent rapidly decreases as we 

 increase s, i.e., as we get back in ancestry the mid-parent comes more- 

 and more nearly to represent in all cases the mean of the general 

 population. Whether the correlation tends to decrease or increase 

 will depend on the relative rates of change of 2& and rp. 



Since p P must always be less than 1, we obtain at once the interest- 

 ing limit that the correlation of an individual and a jpth parent is 

 always less than (0'5)^\ 



For example the correlation between : 



Offspring and parent must be less than 0*71 



„ and grandparents „ „ 0*5 



,, and great-grandparents „ ,, 0*36 



and great-great-grandparents „ „ 0*25 



Their actual values as deduced from Mr. Galton's law are much 

 smaller, as we shall see later. 



(3) The reader will remark that in order to get these results in a 

 simple form we have multiplied the female deviations from the mean 

 by a constant factor m, which has afterwards been taken equal to 

 the ratio of male to female variability. The reason for this was two- 

 fold. In the first place a is certainly not equal to <r', and, conse- 

 quently, m = 1 would not have given 



2. = J;**, but = ^ <V + ff ? * 



a more complex form. In the next place we note the fairly close- 

 equality of r'j r", r"', r"'\ when we neglect reproductive selection j 

 hence m = <j s /<t' s is the only value which appreciably reduces 

 formula (iii) as well as formula (i). I therefore define a mid-parent 

 to be one in which the deviations of the females are reduced to the 

 male standard by first multiplying them by the ratio of male to 



* ' Koy. Soc. Proc.,' vol. 60, p. 278, 



