70 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE 



The total frequency of the array is (1 + )"-/,. Hence, if m 1 , be the mean of the 

 grandchildren measured from the same origin, we have 



/! X (I)"" X m', =/ (1 + fn + i-(n - s) {2 + %(n -- 1)} 



+ / & \ o ( ^ & ) ( n s "" i ) ( o i 5 / ^ o \ / 



it) ^ > & ~r~ 8 ' (n "It 



\bl 10 I I o \ /' 



or 



Thus 



Mean of grandchildren = f n ^ (n s). 



Deviation from general population mean = \n $ (n s) = -^ (4s n). 

 Deviation of grandparent from general population mean = s \n \ (4s n). * 



Hence 



Deviation of offspring _ , 

 Deviation of grandparent 



This ratio is the same whatever he the allogenic constitution of the grandparent. 



(9.) PROPOSITION VII. Tu find the Array of Offspring due to an m th Great-grand- 

 father of s-allogenic Couplets, supposing Complete Random Mating in each 

 Generation. 



The array due to a father of s-allogenic couplets is 



4" X 4" X c v , 6 {%(u + v)}> II (u + v) + (v + w)}"-', 



and, as we have already seen, we must multiply by 4" X 4" and put | (u -j- v) for u, 

 (u + 2v -\- w) for v, and \ (c + w) for w to get the array due to the grandparent of 

 s allogenic units. This process must be repeated m times if we wish to obtain the 

 array due to the -)"' great-grandparent. 



We must first investigate what happens to | (u + v) if this interchange be made m 

 times. Suppose that it has been done * times, and let the answer be 



Kepeat the operation, and the expression becomes 



(f M,- + iM',-) 4 (u + v) + (iM ; + f M',-) | (v + w), 



