408 Prof. Karl Pearson. 



where n is tlie number of pairs of cousins corresponding to h u hi". 

 The factor 2 (see footnote, p. 404) does not occur here, as the cousins 

 form parts of separate, and not identical, arrays. Now let us sum for 

 all possible mid-parental systems, then if r be the correlation of 

 cousins and N" the total number of cousin pairs : 



BTo-oVes S[S(^ 2 )] 



= T V{S(*V') + S[nfc (&i+V)] + S(^ 2 )}. 

 But S(nhjii") = product moment for pairs of brothers = N<r 2 r. 

 S(wA- 2 ) = N<r V, by what precedes. 



S[w/ff (^i + ^i")] ^ s exactly the same as the sum of all offspring 

 with the mid-parental system of ancestry beyond, since hi 

 is not to be equal to hi", 



= 2 S [nhi (\k 2 +!& 3 )] for all values of h x . 



= 2N"(J/9iff 2 1 + J/> a ff 2 2 +-|-/)3ffo23+ . . . . ), 



= 2N<7 Xicx 2 + iccc i + jrCoc e + ....), 



= 2No- 2 - L - = 0'4X No- 2 . 

 1 — fa 



Thus Nff V = T ^ISr ( 7 2 (2r -J-0'4), 

 and r' = 0075. 



Mr. Galton's value is ¥ 2 T = 0*074 ('Natural Inheritance,' p. 133). 

 Had we, however, applied his method correctly, considering cousins 

 as the offspring of brothers, and adopted the value 0*3 given by his 

 law of ancestral heredity for parent and offspring, we should have 

 found 0*0360, instead of our present 0'075. Considering cousins as 

 having two grandparents the same, we should have found 0'0450. 



Second Cousins. — The correlated parts of their mid-parental systems 

 are 



4^-1 + Te" ^2 + T¥ ^Oj 



where hi and h x are cousins, h 2 and h 2 brethren, and 



h = %h+ih+ih+.... 



is the mid-parental system of h 2 and h%. 



In order to work out the correlation, we shall clearly want that of 

 hi and h 2 , or of nephew and uncle. 



Here x x = fco+aj' + 



x 2 = k + x", 



give the correlated parts of the mid-parental systems. 



