CONDUCTING BREEDING INVESTIGATIONS 599 
vidual might have in a given generation, and the amount of repetition 
which has occurred in its ancestry. 
In the above formula for Z, the coefficient of inbreeding, P44 
denotes the maximum possible number of different ancestors which an 
individual might have in matings of the (nm + 1)th generation and 
Q,+1, the actual number of different ancestors which he has. For an 
application of this coefficient we take the pedigree of Roan Gauntlet as 
given in Fig, 232. It gives the following series of values for Z: 
Zo = ee = O per cent. 
VS = = 0 per cent. 
Z, = a = 25 per cent. 
Fo me 10) = 37.5 per cent. 
Fist mew = 40.625 per cent. 
To determine these values we have started with the A, generation in 
which pi = 2 and of necessity gq: = 2 also. The value of Zo, therefore, 
must be 0 in all cases. In the Ae generation p. = 4 and q = 4, also, 
because all these four animals are different and have not previously 
appeared in the pedigree. The value for Zi, therefore, isO0. In genera- 
tion A3, ps = 8. Champion of England appears twice in this genera- 
tion, and since he has appeared already in Ae, the two reappearances 
in A; are crossed out. Counting the remaining individuals in this genera- 
tion, we find g; = 6, and consequently Z; = 25 per cent. Now this Z. = 
25 signifies not only that Roan Gauntlet in the third ancestral generation 
has 25 per cent. less than the maximum possible number of different 
ancestors, but also that in any generation further removed he must of 
necessity have at least 25 per cent. less than the maximum possible 
number of ancestors. In the next following generation, A1, ps = 16. 
In determining g,41 we strike out Lancaster Comet and Virtue, sire 
and dam respectively, of Champion of England. It is worth while noting 
here that these two animals are automatically eliminated in this genera- 
tion because of the reappearance of an animal in a lower generation in this 
same line of descent. Reappearances at the apex of a line of descent are 
called primary reappearances and are marked ® in this pedigree, whereas 
reappearances which are determined by the primary reappearance of 
an individual in a lower generation are called secondary reappearances 
and they are marked with the sign X. It is only necessary to determine 
primary reappearances in calculating the coefficient of inbreeding, for 
secondary reappearances may be accounted for by simply doubling the 
total number of reappearances in the next lower generation. Continuing 
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