INHERITANCE, WITH SPECIAL KEFEKENCE TO MENDEL'S LAWS. 67 



is the mean standard deviation of all the arrays of the first character for a given value 

 of the second. This expression is no longer the actual standard deviation of each 

 array. 



It is of interest to see that this general law of linear regression is verified in the 

 present case. We have cr = x/fV 1 and r = i- Hence if 2 m be the mean standard 

 deviation of the arrays, we should expect 



V 2 _3. n /I 1\ _. ! 



-IB i6 n ( L 97 n - 



Remembering the weight of each array,* we have 



4 X 4" X 4" X 2, 3 = 'I" j 4" X 4c B , 3"- 



= o L 



36 



= 4" X 4" (1 + 3)" 3%n + 4" X 4"n4"- 1 - 3 * 6 - 

 = 4" X 4" X 4" X -J-n, 



whence S,,, 2 = -^n, as we anticipated. 



Corollary (ii.). It is clear that some arrays of offspring will be more, others less 

 variable than the general population. The standard deviation of an array will be 

 equal to that of the general population when s is found from 



-re ($n + 4s) = -? 6 -n, or s = T 7 -n. 



Now the mean number of allogeriic couplets in the general population is \-n. Thus 

 the offspring array equally variable with the general population is at distance -fan 

 from the mean. But cr for the general population = \/-fan. Hence, if we take 

 fathers deviating from the population mean by \/-fan X cr, we should expect their 

 offspring to be equally variable with the general population. Supposing, therefore, 

 the theory under discussion were true, we should have a means of finding, at least 

 approximately, the number n of couplets corresponding to the character under 

 consideration. All we should have to do would be to find the standard deviation of 

 each array of offspring corresponding to a given father ; these standard deviations 

 ought to increase or decrease steadily across the table, their squares giving a straight 

 line when plotted. Smooth the results, interpolate a value equal to cr, and we shall 

 have the character of the father whose offspring are equally variable with the general 

 population ; but the deviation of this father from the mean ought to be <*/-$$n X cr. 

 Hence, since cr can be found, we have at once an approximate value of n. Approxi- 

 mate only, of course, because our arrays are classed by units of measurement, inches, 



* The number of offspring in the array due to the fathers with s-allogenic couplets is found at once 

 by putting u=o-w in the formula of Corollary (i.) on p. 61, and equals 4"x 4 < <, , 3 n ~'. 



K 2 



