406 Prof. Karl Pearson, 



and note that 



where <r is the standard deviation of the offspring, N is the total 

 number of pairs of brothers or mid-parents of each order, and *2q is 

 as before the standard deviation of the group of qth mid-parents. 

 Noticing that pq'^q = coc [ i'~i\ we have if r be the correlation between 

 brothers 



NVr = S[S(^ 2 )] = 2S(»&o 2 ) 



= 2]S" < 7o 2 S(7 2 ^ + 2 7 2 ^+2'ca ( 2 , -e ! ), 



the sum now referring to all values of q and q' from 1 to oc, q being* 

 unequal to q', and q -*» q taken positive. 



Thus : r = 2 7 2 (/3 2 + ficx + /3 4 ca 2 + /3 5 c* 3 + . . . . 



+ ^ + /3 4 + /3 5 ca + y8 6 to 2 + 



+ /3 4 Ca 2 +y8 5 C^ + ^ 6 + y8 7 C^+ 



+ /3 5 ^ 3 + /3 6 ca 2 + /Pcx -f /3 8 + .... 



+ ) 



Hence summing parallel to the diagonal : 



'"<'{i4(' + T3r)} <-*>• 



2o/ 2 



— by (xii), (xiii), and (xvii). 



gry 2 — 1— 27 



Let us evaluate this on Mr. Galton's law and on the hypothesis of 

 a 10 per cent. tax. # On the first hypothesis (3 = — ~, a = -i-, and 



2 y 2 -y/ 2 



7 = 1, hence r = 0'4. On the second hypothesis (p. 403) ft = 0*39284, 

 oc — 0-74639, and 7 = 0'9 ; hence r = 0'4402. 



We can also obtain less accurate values of fraternal correlation in 

 other ways. Suppose two brothers to be considered as sons of one 

 mid-parent hi only. In this case we must take 0"6 for the regression 

 (see the table, p. 403), or 



x x — 0*6 + x% = 0'6 + 



and as before : 



SfoasO = 2 x (0-6) 2 x S(nfc ; 2 ), 

 Mo- V = 2 X 0-36 X N^ 2 , 

 r = 0*36. 



* . [The above value for fraternal correlation shows that 7 must be > 0*6076; 

 that a must be < 1, only gives 7 >0 '5469.] 



