406 Dr. F. Galton. The average Contribution of each 



contributions of T. supplied by knoivn ancestors. Nothing" further 

 need be said about these, except that they are styled coefficients 

 because they must be multiplied into the total number of offspring, 

 in order to calculate the number of them that will, on their separate 

 and independent accounts, be probably tricolours. 



We have next to explain how the coefficients for the unknown 

 ancestry have been calculated, namely, those which are entered in the 

 lower part of Table IV. Suppose all the four grandparents, A 2 , to 

 be tricolour, then only 0*79 of A 3 will be tricolour also, (0*79) 2 of A 4 , 

 and so on. These several orders of ancestry will respectively con- 

 tribute an average of tricolour to each a of the amounts of 

 (0*5) 3 x (0*79), (0*5) 4 x (0-79) 2 , &c. Consequently the sum of their 

 tricolour contributions is 



(0*5) 3 x (0-79) {l + (0-5) x (0*79) + (0*5) 2 x (079) 2 + &c.}, 



which equals 0*1632. The average tricolour contribution from the 

 ancestry of each of the four tricolour grandparents must be reckoned 

 as the quarter of this, namely, 0'0403. 



By a similar process, the average tricolour contribution from the 

 ancestry of each non-tricolour grandparent is found to be 0*0243. 



When the furthest known generation is that of the great-grand- 

 parents, the formula differs from the foregoing only by substituting 

 (0*5) 4 x (0*79) for (0*5) 3 X (0*79). This makes the average tricolour 

 contribution from the ancestry of the whole eight tricolour great- 

 grandparents equal to 0"08160, and that from the ancestry of each of 

 them to be one-eighth of this, or 0*0102. 



In a similar way the tricolour contribution from the ancestry of 

 each non- tricolour great-grandparent is found to be 0'0061. 



The following example shows how the coefficients in Table IV were 

 utilised in calculating the general coefficients entered in Table V. 



2 Parents, T x (personal) 0*5000 



3 Grandparents, T 3 (personal) 0*1875 



1 G-randparent, N 2 (personal) — 



3 Grandparents, T 2 (ancestral). „ .... 0*1224 

 1 Grandparent, N 2 (ancestral) 0*0243 



Total tricolour contribution . . 0*8342 



The coefficient 0*83 will consequently be found under the appro- 

 priate head in Table V, where the total number of offspring ("all 

 cases") is recorded as 119. By multiplying these together, viz., 

 0*83 x 119, the " calculated " number of 99 is obtained. It will be 

 seen that the observed number was 101, a difference of only 2 per 

 cent. 



The extraordinarily close coincidence throughout the two tables, 



